Math lm unit 2

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Mathematics
Department of Education
Republic of the Philippines
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educators from public and private schools, colleges, and/or universities.
We encourage teachers and other education stakeholders to email their
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Learner’s Module
Unit 2
All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means -
electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.

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Mathematics – Grade 10
Learner’s Module
First Edition 2015
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Introduction
This material is written in support of the K to 12 Basic Education
Program to ensure attainment of standards expected of students.
In the design of this Grade 10 materials, it underwent different
processes - development by writers composed of classroom teachers, school
heads, supervisors, specialists from the Department and other institutions;
validation by experts, academicians, and practitioners; revision; content
review and language editing by members of Quality Circle Reviewers; and
finalization with the guidance of the consultants.
There are eight (8) modules in this material.
Module 1 – Sequences
Module 2 – Polynomials and Polynomial Equations
Module 3 – Polynomial Functions
Module 4 – Circles
Module 5 – Plane Coordinate Geometry
Module 6 – Permutations and Combinations
Module 7 – Probability of Compound Events
Module 8 – Measures of Position
With the different activities provided in every module, may you find this
material engaging and challenging as it develops your critical-thinking and
problem-solving skills.

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Unit 2
Module 3: Polynomial Functions............................................................ 99
Lessons and Coverage........................................................................ 100
Module Map......................................................................................... 100
Pre-Assessment .................................................................................. 101
Learning Goals and Targets ................................................................ 105
Activity 1.................................................................................... 106
Activity 2.................................................................................... 107
Activity 3.................................................................................... 108
Activity 4.................................................................................... 108
Activity 5.................................................................................... 110
Activity 6.................................................................................... 111
Activity 7.................................................................................... 112
Activity 8.................................................................................... 115
Activity 9.................................................................................... 115
Activity 10.................................................................................. 118
Activity 11.................................................................................. 119
Activity 12.................................................................................. 121
Activity 13.................................................................................. 122
Activity 14.................................................................................. 123
Summary/Synthesis/Generalization........................................................... 125
Glossary of Terms ...................................................................................... 125
References Used in this Module................................................................. 126
Module 4: Circles .................................................................................... 127
Lessons and Coverage........................................................................ 127
Module Map......................................................................................... 128
Pre-Assessment .................................................................................. 129
Learning Goals and Targets ................................................................ 134
Lesson 1A: Chords, Arcs, and Central Angles.......................................... 135
Activity 1.................................................................................... 135
Activity 2.................................................................................... 137
Activity 3.................................................................................... 138
Activity 4.................................................................................... 139
Activity 5.................................................................................... 150
Activity 6.................................................................................... 151
Activity 7.................................................................................... 151
Activity 8.................................................................................... 152
Activity 9.................................................................................... 152
Activity 10.................................................................................. 155
Activity 11.................................................................................. 155
Activity 12.................................................................................. 157
Activity 13.................................................................................. 159
Table of Contents

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Summary/Synthesis/Generalization ...........................................................160
Lesson 1B: Arcs and Inscribed Angles.......................................................161
Activity 1 ....................................................................................161
Activity 2 ....................................................................................162
Activity 3 ....................................................................................163
Activity 4 ....................................................................................164
Activity 5 ....................................................................................167
Activity 6 ....................................................................................168
Activity 7 ....................................................................................169
Activity 8 ....................................................................................170
Activity 9 ....................................................................................172
Activity 10 ..................................................................................174
Activity 11 ..................................................................................175
Activity 12 ..................................................................................176
Lesson 2A: Tangents and Secants of a Circle ............................................178
Activity 1 ....................................................................................178
Activity 2 ....................................................................................179
Activity 3 ....................................................................................180
Activity 4 ....................................................................................188
Activity 5 ....................................................................................189
Activity 6 ....................................................................................192
Activity 7 ....................................................................................194
Activity 8 ....................................................................................197
Lesson 2B: Tangent and Secant Segments.................................................199
Activity 1 ....................................................................................199
Activity 2 ....................................................................................200
Activity 3 ....................................................................................200
Activity 4 ....................................................................................201
Activity 5 ....................................................................................204
Activity 6 ....................................................................................205
Activity 7 ....................................................................................206
Activity 8 ....................................................................................207
Activity 9 ....................................................................................208
Activity 10 ..................................................................................210
Glossary of Terms.......................................................................................212
List of Theorems and Postulates on Circles...............................................213
References and Website Links Used in this Module..................................215
Module 5: Plane Coordinate Geometry ...............................................221
Lessons and Coverage ........................................................................222
Module Map .........................................................................................222
Pre-Assessment...................................................................................223
Learning Goals and Targets.................................................................228

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Lesson 1: The Distance Formula, the Midpoint Formula,
and the Coordinate Proof .......................................................... 229
Activity 1.................................................................................... 229
Activity 2.................................................................................... 230
Activity 3.................................................................................... 231
Activity 4.................................................................................... 232
Activity 5.................................................................................... 241
Activity 6.................................................................................... 242
Activity 7.................................................................................... 242
Activity 8.................................................................................... 243
Activity 9.................................................................................... 245
Activity 10.................................................................................. 248
Activity 11.................................................................................. 250
Lesson 2: The Equation of a Circle............................................................ 252
Activity 1.................................................................................... 252
Activity 2.................................................................................... 253
Activity 3.................................................................................... 254
Activity 4.................................................................................... 263
Activity 5.................................................................................... 265
Activity 6.................................................................................... 265
Activity 7.................................................................................... 266
Activity 8.................................................................................... 267
Activity 9.................................................................................... 267
Activity 10.................................................................................. 269
Glossary of Terms ...................................................................................... 270
References and Website Links Used in this Module ................................. 271

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I. INTRODUCTION
You are now in Grade 10, your last year in junior high school. In
this level and in the higher levels of your education, you might ask the
question: What are math problems and solutions for? An incoming college
student may ask, “How can designers and manufacturers make boxes
having the largest volume with the least cost?” And anybody may ask: In
what other fields are the mathematical concepts like functions used? How
are these concepts applied?
Look at the mosaic picture below. Can you see some mathematical
representations here? Give some.
As you go through this module, you are expected to define and
illustrate polynomial functions, draw the graphs of polynomial functions
and solve problems involving polynomial functions. The ultimate goal of
this module is for you to answer these questions: How are polynomial
functions related to other fields of study? How are these used in solving
real-life problems and in decision making?

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II. LESSON AND COVERAGE
This is a one-lesson module. In this module, you will learn to:
 illustrate polynomial functions
 graph polynomial functions
 solve problems involving polynomial functions
Solutions of Problems Involving
Polynomial Functions
Graphs of Polynomial
Functions
Illustrations of Polynomial
Functions
The Polynomial Functions

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III. PRE-ASSESSMENT
Part 1
Let us find out first what you already know related to the content of this
module. Answer all items. Choose the letter that best answers each
question. Please take note of the items/questions that you will not be able
to answer correctly and revisit them as you go through this module for
self-assessment.
1. What should n be if f(x) = xn
defines a polynomial function?
A. an integer C. any number
B. a nonnegative integer D. any number except 0
2. Which of the following is an example of a polynomial function?
A. 13
4
)( 3
 x
x
xf C. 6
27)( xxxf 
B. 22
3
2
3
2)( xxxf  D. 53)( 3
 xxxf
3. What is the leading coefficient of the polynomial function
42)( 3
 xxxf ?
A. 1 C. 3
B. 2 D. 4
4. How should the polynomial function 432)( 53
 xxxxf be written
in standard form?
A. 432)( 53
 xxxxf C. 53
324)( xxxxf 
B. 35
234)( xxxxf  D. 423)( 35
 xxxxf
5. Which of the following could be the graph of the polynomial function
1234 23
 xxxy ?
A. B. C. D.
x x
x
yy y
y

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6. From the choices, which polynomial function in factored form
represents the given graph?
7. If you will draw the graph of 2
)2(  xxy , how will you sketch it with
respect to the x-axis?
A. Sketch it crossing both (-2,0) and (0,0).
B. Sketch it crossing (-2,0) and tangent at (0,0).
C. Sketch it tangent at (-2,0) and crossing (0,0).
D. Sketch it tangent at both (-2,0) and (0,0).
8. What are the end behaviors of the graph of 432)( 53
 xxxxf ?
A. rises to the left and falls to the right
B. falls to the left and rises to the right
C. rises to both directions
D. falls to both directions
9. You are asked to illustrate the sketch of 43)( 53
 xxxf using its
properties. Which will be your sketch?
A. B. C. D.
A. )1)(1)(2(  xxxy
B. )2)(1)(1(  xxxy
C. )1)(1)(2(  xxxxy
D. )2)(1)(1(  xxxxy
y
x
y
x
y
x
y
x

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10. Your classmate Linus encounters difficulties in showing a sketch of
the graph of 3 2
2 3 4y x x x    6. You know that the quickest
technique is the Leading Coefficient Test. You want to help Linus in
his problem. What hint/clue should you give?
A. The graph falls to the left and rises to the right.
B. The graph rises to both left and right.
C. The graph rises to the left and falls to the right.
D. The graph falls to both left and right.
11. If you will be asked to choose from -2, 2, 3, and 4, what values for a
and n will you consider so that y = axn
could define the graph below?
12. A car manufacturer determines that its profit, P, in thousands of
pesos, can be modeled by the function P(x) = 0.00125x4
+ x – 3,
where x represents the number of cars sold. What is the profit when
x = 300?
A. Php 101.25 C. Php 3,000,000.00
B. Php 1,039,500.00 D. Php 10,125,297.00
13. A demographer predicts that the population, P, of a town t years from
now can be modeled by the function P(t) = 6t4
– 5t3
+ 200t + 12 000.
What will the population of the town be two (2) years from now?
A. 12 456 C. 1 245 600
B. 124 560 D. 12 456 000
14. Consider this Revenue-Advertising Expense situation:
The total revenue R (in millions of pesos) for a company is related to
its advertising expense by the function
 3 21
R x 600x , 0 x 400
100 000
    
where x is the amount spent on advertising (in ten thousands of
pesos).
A. a = 2 , n = 3
B. a = 3 , n = 2
C. a = - 2 , n = 4
D. a = - 2 , n = 3

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Currently, the company spends Php 2,000,000.00 for advertisement.
If you are the company manager, what best decision can you make
with this business circumstance based on the given function with its
restricted domain?
A. I will increase my advertising expenses to Php 2,500,000.00
because this will give a higher revenue than what the company
currently earns.
B. I will decrease my advertising expenses to Php 1,500,000.00
because this will give a higher revenue than what the company
currently earns.
C. I will decrease my advertising expenses to Php 1,500,000.00
because lower cost means higher revenue.
D. It does not matter how much I spend for advertisement, my
revenue will stay the same.
Part 2
Read and analyze the situation below. Then, answer the question and
perform the tasks that follow.
Karl Benedic, the president of Mathematics Club, proposed a project:
to put up a rectangular Math Garden whose lot perimeter is 36 meters. He
was soliciting suggestions from the members for feasible dimensions of the
lot.
Suppose you are a member of the club, what will you suggest to Karl
Benedic if you want a maximum lot area? You must convince him through a
mathematical solution.
Consider the following guidelines:
1. Make an illustration of the lot with the needed labels.
2. Solve the problem. Hint: Consider the formulas P = 2l + 2w for
perimeter and A = lw for the area of the rectangle. Use the formula for
P and the given information in the problem to express A in terms of
either l or w.
3. Make a second illustration that satisfies the findings in the solution
made in number 2.
4. Submit your solution on a sheet of paper with recommendations.

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Rubric for Rating the Output:
Score Descriptors
4
The problem is correctly modeled with a quadratic function,
appropriate mathematical concepts are fully used in the
solution, and the correct final answer is obtained.
3
The problem is correctly modeled with a quadratic function,
appropriate mathematical concepts are partially used in the
2
The problem is not properly modeled with a quadratic function,
other alternative mathematical concepts are used in the
1
The problem is not totally modeled with a quadratic function, a
solution is presented but has incorrect final answer.
The additional two (2) points will be determined from the illustrations
made. One (1) point for each if properly drawn with necessary labels.
IV. LEARNING GOALS AND TARGETS
After going through this module, you should be able to demonstrate
understanding of key concepts on polynomial functions. Furthermore, you
should be able to conduct a mathematical investigation involving
polynomial functions.

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Start this module by recalling your knowledge on the concept of
polynomial expressions. This knowledge will help you understand the
formal definition of a polynomial function.
Determine whether each of the following is a polynomial expression or not.
Give your reasons.
1. 14x 6. 
2. 3
5 4 2x x x  7. 3 2
3 3 9 2x x x  
3. 2014x 8. 3
2 1x x 
4.
3 1
4 4
3 7x x  9. 100 100
4 4x x
 
5. 3 4 5
1 2 3
2 3 4x x x
  10. 1 – 16x2
Did you answer each item correctly? Do you remember when an
expression is a polynomial? We defined a related concept below.
A polynomial function is a function of the form
   
 
     1 2
1 2 1 0
...n n n
n n n
P x a x a x a x a x a , ,0na
where n is a nonnegative integer, naaa ...,,, 10 are real numbers called
coefficients, n
n xa is the leading term, na is the leading coefficient,
and 0a is the constant term.
The terms of a polynomial may be written in any order. However, if
they are written in decreasing powers of x, we say the polynomial function is
in standard form.
Other than P(x), a polynomial function may also be denoted by f(x).
Sometimes, a polynomial function is represented by a set P of ordered pairs
(x,y). Thus, a polynomial function can be written in different ways, like the
following.
Activity 1:

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f ( x )  
 
     1 2
1 2 1 0
...n n n
n n n
a x a x a x a x a
or
1 2
1 2 1 0
...n n n
n n n
y a x a x a x a x a 
 
     
Polynomials may also be written in factored form and as a product of
irreducible factors, that is, a factor that can no longer be factored using
coefficients that are real numbers. Here are some examples.
a. y = x4
+ 2x3
– x2
+ 14x – 56 in factored form is y = (x2
+ 7)(x – 2)(x + 4)
b. y = x4
+ 2x3
– 13x2
– 10x in factored form is y = x(x – 5)(x + 1)(x + 2)
c. y = 6x3
+ 45x2
+ 66x – 45 in factored form is y = 3(2x – 1)(x + 3)(x + 5)
d. f(x) = x3
+ x2
+ 18 in factored form is f(x) = (x2
– 2x + 6)(x + 3)
e. f(x) = 2x3
+ 5x2
+ 7x – 5 in factored form is f(x) = (x2
+ 3x + 5)(2x – 1)
Consider the given polynomial functions and fill in the table below.
Polynomial Function
Polynomial
Function in
Standard Form
Degree
Leading
Coefficient
Constant
Term
1. f ( x ) = 2 – 11x + 2x2
2. f ( x )  
3
2 5
15
3 3
x
x
3. y = x (x2
– 5)
4.   3 3y x x x  
5. 2
)1)(1)(4(  xxxy
After doing this activity, it is expected that the definition of a polynomial
function and the concepts associated with it become clear to you. Do the
next activity so that your skills will be honed as you give more examples of
Activity 2:

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Use all the numbers in the box once as coefficients or exponents to form as
many polynomial functions of x as you can. Write your polynomial functions in
standard form.
1 –2
4
7
2
6
1
 3
How many polynomial functions were you able to give? Classify each
according to its degree. Also, identify the leading coefficient and the constant
term.
In this section, you need to revisit the lessons and your knowledge
on evaluating polynomials, factoring polynomials, solving polynomial
equations, and graphing by point-plotting. Your knowledge of these
topics will help you sketch the graph of polynomial functions manually.
You may also use graphing utilities/tools in order to have a clearer view
and a more convenient way of describing the features of the graph. Also,
you will focus on polynomial functions of degree 3 and higher, since
graphing linear and quadratic functions were already taught in previous
grade levels. Learning to graph polynomial functions requires your
appreciation of its behavior and other properties.
Factor each polynomial completely using any method. Enjoy working with
your seatmate using the Think-Pair-Share strategy.
1. (x – 1) (x2
– 5x + 6)
2. (x2
+ x – 6) (x2
– 6x + 9)
3. (2x2
– 5x + 3) (x – 3)
4. x3
+ 3x2
– 4x – 12
5. 2x4
+ 7x3
– 4x2
– 27x – 18
Did you get the answers correctly? What method(s) did you use? Now,
do the same with polynomial functions. Write each of the following polynomial
functions in factored form:
Activity 4:
Activity 3:

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6. xxxy 1223

7. 164
 xy
8. 68482 234
 xxxxy
9. xxxy 910 35

10. 1827472 234
 xxxxy
The preceding task is very important for you since it has something to
do with the x-intercepts of a graph. These are the x-values when y = 0,
thus, the point(s) where the graph intersects the x-axis can be determined.
To recall the relationship between factors and x-intercepts, consider
these examples:
a. Find the intercepts of 64 23
 xxxy .
Solution:
To find the x-intercept/s, set y = 0. Use the factored form. That is,
y = x3
– 4x2
+ x + 6
y = (x + 1)(x – 2)(x – 3) Factor completely.
0 = (x + 1)(x – 2)(x – 3) Equate y to 0.
x + 1 = 0
x = –1
or x – 2 = 0
x = 2
or x – 3 = 0
x = 3
The x-intercepts are –1, 2, and 3. This means the graph will pass
through (-1, 0), (2, 0), and (3, 0).
Finding the y-intercept is more straightforward. Simply set x = 0
in the given polynomial. That is,
y = x3
– 4x2
+ x + 6
y = 03
– 4(0)2
+ 0 + 6
y = 6
The y-intercept is 6. This means the graph will also pass through (0,6).
Equate each factor to 0
to determine x.

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b. Find the intercepts of xxxxy 66 234

Solution:
For the x-intercept(s), find x when y = 0. Use the factored form.
That is,
y = x4
+ 6x3
– x2
– 6x
y = x(x + 6)(x + 1)(x – 1)
0 = x(x + 6)(x + 1)(x – 1)
x=0 or x+6 =0
x=–6
or x+1 =0
x=–1
or x–1=0
x=1
The x-intercepts are -6, -1, 0, and 1. This means the graph will pass
through (-6,0), (-1,0), (0,0), and (1,0).
Again, finding the y-intercept simply requires us to set x = 0 in
the given polynomial. That is,
4 3 2
6 6y x x x x   
y  (04
)  6(03
) (02
) 6(0)
0y 
The y-intercept is 0. This means the graph will pass also through (0,0).
You have been provided illustrative examples of solving for the x- and
y- intercepts, an important step in graphing a polynomial function. Remember,
these intercepts are used to determine the points where the graph intersects
or touches the x-axis and the y-axis. But these points are not sufficient to
draw the graph of polynomial functions. Enjoy as you learn by performing the
next activities.
Determine the intercepts of the graphs of the following polynomial functions:
1. y = x3
+ x2
– 12x
2. y = (x – 2)(x – 1)(x + 3)
3. y = 2x4
+ 8x3
+ 4x2
– 8x – 6
4. y = –x4
+ 16
5. y = x5
+ 10x3
– 9x
You have learned how to find the intercepts of a polynomial function.
You will discover more properties as you go through the next activities.
Activity 5:
Equate each factor to
0 to determine x.
Factor completely.
Equate y to 0.

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Work with your friends. Determine the x-intercept/s and the y-intercept of
each given polynomial function. To obtain other points on the graph, find the
value of y that corresponds to each value of x in the table.
1. y = (x + 4)(x + 2)(x – 1)(x – 3) x-intercepts: __ __ __ __
y-intercept: __
x -5 -3 0 2 4
y
List all your answers above as ordered pairs.
2. y = –(x + 5)(2x + 3)(x – 2)(x – 4) x-intercepts: __ __ __ __
y-intercept: __
x -6 -4 -0.5 3 5
y
y = –x(x + 6)(3x – 4) x-intercepts: __ __ __
y-intercept: __
x -7 -3 1 2
y
3. y = x2
(x + 3)(x + 1)(x – 1)(x – 3) x-intercepts: __ __ __ __ __
y-intercept: __
x -4 -2 -0.5 0.5 2 4
y
List all your answers above as ordered pairs
In this activity, you evaluated a function at given values of x. Notice that
some of the given x-values are less than the least x-intercept, some are between
two x-intercepts, and some are greater than the greatest x  intercept. For
example, in number 1, the x-intercepts are -4, -2, 1, and 3. The value -5 is
used as x-value less than -4; -3, 0, and 2 are between two x-intercepts; and 4
is used as x-value greater than 3. Why do you think we should consider
them?
Activity 6:

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In the next activity, you will describe the behavior of the graph of a
polynomial function relative to the x-axis.
Given the polynomial function y = (x + 4)(x + 2)(x – 1)(x – 3), complete the
table below. Answer the questions that follow.
Value of
x
Value of
y
Relation of y value to 0:
y > 0, y = 0, or y < 0?
Location of the point (x, y):
above the x-axis, on the
x-axis, or below the x-axis?
-5 144 0y above the x-axis
-4
-3
-2 0 y = 0 on the x - axis
0
1
2
3
4
Questions:
1. At what point(s) does the graph pass through the x-axis?
2. If 4x , what can you say about the graph?
3. If 24  x , what can you say about the graph?
4. If 12  x , what can you say about the graph?
5. If 31  x , what can you say about the graph?
6. If 3x , what can you say about the graph?
Now, this table may be transformed into a simpler one that will instantly
help you in locating the curve. We call this the table of signs.
The roots of the polynomial function y = (x + 4)(x + 2)(x – 1)(x – 3) are
x = –4, –2, 1, and 3. These are the only values of x where the graph will cross
the x-axis. These roots partition the number line into intervals. Test values are
then chosen from within each interval.
Activity 7:

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The table of signs and the rough sketch of the graph of this function
can now be constructed, as shown below.
The Table of Signs
Intervals
4x   4 2x    2 1x   1 3x  3x 
Test value -5 -3 0 2 4
4x  – + + + +
2x  – – + + +
1x  – – – + +
3x  – – – – +
)3)(1)(2)(4(  xxxxy + – + – +
Position of the curve
relative to the x-axis
above below above below above
The Graph of )3)(1)(2)(4(  xxxxy
We can now use the information from the table of signs to construct a
possible graph of the function. At this level, though, we cannot determine the
turning points of the graph, we can only be certain that the graph is correct
with respect to intervals where the graph is above, below, or on the x-axis.
The arrow heads at both ends of the graph signify that the graph
indefinitely goes upward.

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Here is another example: Sketch the graph of )43)(2()(  xxxxf
Roots of f(x): -2, 0,
3
4
Table of Signs:
Intervals
2x   2 0x  
4
0
3
x 
4
3
x 
Test value -3 -1 1 2
–x + + – –
x + 2 – + + +
3x – 4 – – – +
f(x) = –x(x + 2)(3x – 4) + – + –
Position of the curve
relative to the x-axis
above below above below
Graph:
In this activity, you learned how to sketch the graph of polynomial
functions using the intercepts, some points, and the position of the curves
determined from the table of signs. The procedures described are applicable
when the polynomial function is in factored form. Otherwise, you need to
express first a polynomial in factored form. Try this in the next activity.

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For each of the following functions, give
(a) the x-intercept(s)
(b) the intervals obtained when the x-intercepts
are used to partition the number line
(c) the table of signs
(d) a sketch of the graph
1. y = (2x + 3)(x – 1)(x – 4)
2. y = –x3
+ 2x2
+ 11x – 2
3. y = x4
– 26x2
+ 25
4. y = –x4
– 5x3
+ 3x2
+ 13x – 10
5. y = x2
(x + 3)(x + 1)4
(x – 1)3
Post your answer/output for a walk-through. For each of these
polynomial functions, answer the following:
a. What happens to the graph as x decreases without bound?
b. For which interval(s) is the graph (i) above and (ii) below the
x-axis?
c. What happens to the graph as x increases without bound?
d. What is the leading term of the polynomial function?
e. What are the leading coefficient and the degree of the function?
Now, the big question for you is: Do the leading coefficient and degree
affect the behavior of its graph? You will answer this after an investigation in
the next activity.
After sketching manually the graphs of the five functions given in Activity 8,
you will now be shown polynomial functions and their corresponding graphs.
Study each figure and answer the questions that follow. Summarize your
answers using a table similar to the one provided.
Activity 9:
Activity 8:

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Case 1
The graph on the right is defined by
y = 2x3
– 7x2
– 7x + 12
or, in factored form,
y = (2x + 3) (x – 1) (x – 4).
Questions:
a. Is the leading coefficient a positive or a
negative number?
b. Is the polynomial of even degree or odd
degree?
c. Observe the end behaviors of the graph on
both sides. Is it rising or falling to the left or to
the right?
Case 2
473 2345
 xxxxy
22
)2)(1()1(  xxxy .
Questions:
negative number?
degree?
c. Observe the end behaviors of the graph
on both sides. Is it rising or falling to the
left or to the right?
Case 3
xxxy 67 24
 or, in factored form,
)2)(1)(3(  xxxxy .
Questions:
negative number?
degree?
x
y
x
y
x
y

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Case 4
2414132 234
 xxxxy
)4)(2)(1)(3(  xxxxy .
Questions:
negative number?
degree?
Now, complete this table. In the last column, draw a possible graph for
the function, showing how the function behaves. (You do not need to place
your graph on the xy-plane). The first one is done for you.
Sample Polynomial Function
Leading
Coefficient:
0n or
0n
Degree:
Even or
Odd
Behavior of the
Graph:
Rising or Falling
Possible
Sketch
Left-
hand
Right-
hand
1. 3 2
2 7 7 12y x x x    0n odd falling rising
2. 5 4 3 2
3 7 4y x x x x    
3. 4 2
7 6y x x x  
4. 4 3 2
2 13 14 24y x x x x    
Summarize your findings from the four cases above. What do you observe
if:
1. the degree of the polynomial is odd and the leading coefficient is
positive?
2. the degree of the polynomial is odd and the leading coefficient is
negative?
3. the degree of the polynomial is even and the leading coefficient is
positive?
4. the degree of the polynomial is even and the leading coefficient is
negative?
x
y

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Congratulations! You have now illustrated The Leading Coefficient
Test. You should have realized that this test can help you determine the end
behaviors of the graph of a polynomial function as x increases or decreases
without bound.
Recall that you have already learned two properties of the graph of
polynomial functions; namely, the intercepts which can be obtained from the
Rational Root Theorem, and the end behaviors which can be identified using
the Leading Coefficient Test. Another helpful strategy is to determine whether
the graph crosses or is tangent to the x-axis at each x-intercept. This strategy
involves the concept of multiplicity of a zero of a polynomial function.
Multiplicity tells how many times a particular number is a zero or root for the
given polynomial.
The next activity will help you understand the relationship between
multiplicity of a root and whether a graph crosses or is tangent to the x-axis.
Given the function )2()1()1()2( 432
 xxxxy and its graph, complete the
table below, then answer the questions that follow.
Root or Zero Multiplicity
Characteristic
of Multiplicity:
Odd or Even
Behavior of Graph
Relative to x-axis at this
Root:
Crosses or Is Tangent to
-2
-1
1
2
Activity 10:
x
y

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Questions:
a. What do you notice about the graph when it passes through a root of
even multiplicity?
b. What do you notice about the graph when it passes through a root of
odd multiplicity?
This activity extends what you learned when using a table of signs to
graph a polynomial function. When the graph crosses the x-axis, it means the
graph changes from positive to negative or vice versa. But if the graph is
tangent to the x-axis, it means that the graph is either positive on both sides
of the root, or negative on both sides of the root.
In the next activity, you will consider the number of turning points of the
graph of a polynomial function. The turning points of a graph occur when the
function changes from decreasing to increasing or from increasing to
decreasing values.
Complete the table below. Then answer the questions that follow.
Polynomial Function Sketch Degree
Number of
Turning
Points
1. 4
xy 
2. 152 24
 xxy
3. 5
xy 
Activity 11:
x
y
x
y
x
y

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Polynomial Function Sketch Degree
Number of
Turning
Points
4. 1235
 xxxy
5. xxxy 45 35

Questions:
a. What do you notice about the number of turning points of the quartic
functions (numbers 1 and 2)? How about of quintic functions (numbers
3 to 5)?
b. From the given examples, do you think it is possible for the degree of a
function to be less than the number of turning points?
c. State the relation of the number of turning points of a function with its
degree n.
In this section, you have encountered important concepts that can help
you graph polynomial functions. Notice that the graph of a polynomial
function is continuous, smooth, and has rounded turns. Further, the
number of turning points in the graph of a polynomial is strictly less than the
degree of the polynomial.
Use what you have learned as you perform the activities in the
succeeding sections.
y
x
x
y

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The goal of this section is to help you think critically and creatively
as you apply the techniques in graphing polynomial functions. Also, this
section aims to provide opportunities to solve real-life problems involving
For each given polynomial function, describe or determine the following, then
sketch the graph. You may need a calculator in some computations.
a. leading term
b. end behaviors
c. x-intercepts
points on the x-axis
d. multiplicity of roots
e. y-intercept
point on the y-axis
f. number of turning points
g. sketch
1. )52()1)(3( 2
 xxxy
2. 322
)2()1)(5(  xxxy
3. 422 23
 xxxy
4. )32)(7( 22
 xxxy
5. 2861832 234
 xxxxy
In this activity, you were given the opportunity to sketch the graph of
polynomial functions. Were you able to apply all the necessary concepts and
properties in graphing each function? The next activity will let you see the
connections of these mathematics concepts to real life.
Activity 12:

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Work in groups. Apply the concepts of polynomial functions to answer the
questions in each problem. Use a calculator when needed.
1. Look at the pictures below. What do these tell us? Filipinos need to take
the problem of deforestation seriously.
The table below shows the forest cover of the Philippines in relation to
its total land area of approximately 30 million hectares.
Year 1900 1920 1960 1970 1987 1998
Forest Cover (%) 70 60 40 34 23.7 22.2
Source: Environmental Science for Social Change, Decline of the Philippine Forest
A cubic polynomial that best models the data is given by
3 2
26 3500 391 300 69 717000
; 0 98
1 000 000
x x x
y x
  
 
where y is the percent forest cover x years from 1900.
10 20 30 40 50 60 70 80 90 x
-10
10
20
30
40
50
60
70
80
y
O
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Questions/Tasks:
a. Using the graph, what is the approximate forest cover during the year
1940?
b. Compare the forest cover in 1987 (as given in the table) to the forest
cover given by the polynomial function. Why are these values not
exactly the same?
c. Do you think you can use the polynomial to predict the forest cover in
the year 2100? Why or why not?
2. The members of a group of packaging designers of a gift shop are looking
for a precise procedure to make an open rectangular box with a volume of
560 cubic inches from a 24-inch by 18-inch rectangular piece of material.
The main problem is how to identify the side of identical squares to be cut
from the four corners of the rectangular sheet so that such box can be
made.
Question/Task:
Suppose you are chosen as the leader and you are tasked to lead
in solving the problem. What will you do to meet the specifications needed
for the box? Show a mathematical solution.
Were you surprised that polynomial functions have real and practical
uses? What do you need to solve these kinds of problems? Enjoy learning as
you proceed to the next section.
The goal of this section is to check if you can apply polynomial
functions to real-life problems and produce a concrete object that
satisfies the conditions given in the problem.
Read the problem carefully and answer the questions that follow.
You are designing candle-making kits.
Each kit contains 25 cubic inches of candle wax
and a mold for making a pyramid-shaped candle
with a square base. You want the height of the
candle to be 2 inches less than the edge of the
base.
Activity 14:

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Questions/Tasks:
1. What should the dimensions of your candle mold be? Show a
mathematical procedure in determining the dimensions.
2. Use a sheet of cardboard as sample material in preparing a candle
mold with such dimensions. The bottom of the mold should be closed.
The height of one face of the pyramid should be indicated.
3. Write your solution in one of the faces of your output (mold).
Rubric for the Mathematical Solution
Point Descriptor
4
The problem is correctly modeled with a polynomial function,
appropriate mathematical concepts are used in the solution,
and the correct final answer is obtained.
3
The problem is correctly modeled with a polynomial function,
appropriate mathematical concepts are partially used in the
2
The problem is not properly modeled with a polynomial
function, other alternative mathematical concepts are used in
the solution, and the correct final answer is obtained.
1
The problem is not properly modeled with a polynomial
function, a solution is presented but the final answer is
incorrect.
Criteria for Rating the Output:
 The mold has the needed dimensions and parts.
 The mold is properly labeled with the required length of parts.
 The mold is durable.
 The mold is neat and presentable.
Point/s to Be Given:
4 points if all items in the criteria are evident
3 points if any three of the items are evident
2 points if any two of the items are evident
1 point if any of the items is evident

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SUMMARY/SYNTHESIS/GENERALIZATION
This lesson was about polynomial functions. You learned how to:
 illustrate and describe polynomial functions;
 show the graph of polynomial functions using the following properties:
- the intercepts (x-intercept and y-intercept);
- the behavior of the graph using the Leading Coefficient Test, table
of signs, turning points, and multiplicity of zeros; and
 solve real-life problems that can be modeled with polynomial functions.
GLOSSARY OF TERMS
Constant Function - a polynomial function whose degree is 0
Evaluating a Polynomial - a process of finding the value of the polynomial at
a given value in its domain
Intercepts of a Graph - points on the graph that have zero as either the x-
coordinate or the y-coordinate
Irreducible Factor - a factor that can no longer be factored using coefficients
that are real numbers
Leading Coefficient Test - a test that uses the leading term of the
polynomial function to determine the right-hand and the left-hand behaviors of
the graph
Linear Function - a polynomial function whose degree is 1
Multiplicity of a Root - tells how many times a particular number is a root for
the given polynomial
Nonnegative Integer - zero or any positive integer
Polynomial Function - a function denoted by
01
2
2
1
1 ...)( axaxaxaxaxP n
n
n
n
n
n  


 , where n is a nonnegative
integer, naaa ...,,, 10 are real numbers called coefficients but ,0na n
n xa is
the leading term, na is the leading coefficient, and 0a is the constant term
Polynomial in Standard Form - any polynomial whose terms are arranged in
decreasing powers of x
Quadratic Function – a polynomial function whose degree is 2

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Quartic Function – a polynomial function whose degree is 4
Quintic Function – a polynomial function whose degree is 5
Turning Point – a point where the function changes from decreasing to
increasing or from increasing to decreasing values
REFERENCES USED IN THIS MODULE:
Alferez, M.S., Duro, MC.A. & Tupaz, KK.L. (2008). MSA Advanced Algebra.
Quezon City, Philippines: MSA Publishing House.
Berry, J., Graham, T., Sharp, J. & Berry, E. (2003). Schaum’s A-Z
Mathematics. London, United Kingdom: Hodder & Stoughton
Educational.
Cabral, E. A., De Lara-Tuprio, E.P., De Las Penas, ML. N., Francisco, F. F.,
Garces, IJ. L., Marcelo, R.M. & Sarmiento, J. F. (2010). Precalculus.
Quezon City, Philippines: Ateneo de Manila University Press.
Jose-Dilao, S., Orines, F. B. & Bernabe, J.G. (2003). Advanced Algebra,
Trigonometry and Statistics. Quezon City, Philippines: JTW
Corporation.
Lamayo, F. C. & Deauna, M. C. (1990). Fourth Year Integrated Mathematics.
Quezon City, Philippines: Phoenix Publishing House, Inc.
Larson, R. & Hostetler, R. P. (2012). Algebra and Trigonometry. Pasig City,
Philippines: Cengage Learning Asia Pte. Ltd.
Marasigan, J. A., Coronel, A.C. & Coronel, I.C. (2004). Advanced Algebra
with Trigonometry and Statistics. Makati City, Philippines: The
Bookmark, Inc.
Quimpo, N. F. (2005). A Course in Freshman Algebra. Quezon City,
Philippines.
Uy, F. B. & Ocampo, J.L. (2000). Board Primer in Mathematics. Mandaluyong
City, Philippines: Capitol Publishing House.
Villaluna, T. T. & Van Zandt, GE. L. (2009). Hands-on, Minds-on Activities in
Mathematics IV. Quezon City, Philippines: St. Jude Thaddeus
Publications.

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I. INTRODUCTION
Have you imagined yourself pushing a cart or riding in a bus having
wheels that are not round? Do you think you can move heavy objects from
one place to another easily or travel distant places as fast as you can?
What difficulty do you think would you experience without circles? Have
you ever thought of the importance of circles in the field of transportation,
industries, sports, navigation, carpentry, and in your daily life?
Find out the answers to these questions and determine the vast
applications of circles through this module.
II. LESSONS AND COVERAGE:
In this module, you will examine the above questions when you
take the following lessons:
Lesson 1A – Chords, Arcs, and Central Angles
Lesson 1B – Arcs and Inscribed Angles
Lesson 2A – Tangents and Secants of a Circle
Lesson 2B – Tangent and Secant Segments

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In these lessons, you will learn to:
Lesson 1A
Lesson 1B
 derive inductively the relations among chords, arcs,
central angles, and inscribed angles;
 illustrate segments and sectors of circles;
 prove theorems related to chords, arcs, central angles,
and inscribed angles; and
 solve problems involving chords, arcs, central angles,
and inscribed angles of circles.
Lesson 2A
Lesson 2B
 illustrate tangents and secants of circles;
 prove theorems on tangents and secants; and
 solve problems involving tangents and secants of
circles.
Here is a simple map of the lessons that will be covered in this module:
Circles
Applications of
Circles
Relationships among
Chords, Arcs, Central
Angles, and Inscribed
Angles
Tangents and
Secants of Circles
Chords, Arcs,
and Central
Angles
Arcs and
Inscribed
Angles
Tangents and
Secants
Tangent and
Secant
Segments

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III. PRE-ASSESSMENT
Part I
Find out how much you already know about the topics in this module.
Choose the letter that you think best answers each of the following
questions. Take note of the items that you were not able to answer
correctly and find the right answer as you go through this module.
1. What is an angle whose vertex is on a circle and whose sides contain
chords of the circle?
A. central angle C. circumscribed angle
B. inscribed angle D. intercepted angle
2. An arc of a circle measures 30°. If the radius of the circle is 5 cm, what
is the length of the arc?
A. 2.62 cm B. 2.3 cm C. 1.86 cm D. 1.5 cm
3. Using the figure below, which of the following is an external secant
segment of M?
A. CO C. NO
B. TI D. NI
4. The opposite angles of a quadrilateral inscribed in a circle are _____.
A. right C. complementary
B. obtuse D. supplementary
5. In S at the right, what is VSIm if mVI = 140?
A. 35 C. 140
B. 75 D. 230
6. What is the sum of the measures of the central angles of a circle with
no common interior points?
A. 120 B. 240 C. 360 D. 480
I N
T
E
C
O
M

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P
X
Y
N
M
7. Catherine designed a pendant. It is a regular hexagon set in a circle.
Suppose the opposite vertices are connected by line segments and
meet at the center of the circle. What is the measure of each angle
formed at the center?
A. 5.22 B. 45 C. 60 D. 72
8. If an inscribed angle of a circle intercepts a semicircle, then the angle
is _________.
A. acute B. right C. obtuse D. straight
9. At a given point on the circle, how many line/s can be drawn that is
tangent to the circle?
A. one B. two C. three D. four
10. What is the length of ZK in the figure on the right?
A. 2.86 units C. 8 units
B. 6 units D. 8.75 units
11. In the figure on the right, mXY = 150 and mMN = 30.
What is XPYm ?
A. 60
B. 90
C. 120
D. 180
12. The top view of a circular table shown on the
right has a radius of 120 cm. Find the area of
the smaller segment of the table (shaded
region) determined by a 60 arc.
A.  2400 3600 3  cm2
B. 3600 3 cm2
C. 2400 cm2
D.  14 400 3600 3  cm2
60°
120 cm

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13. In O given below, what is PR if NO = 15 units and ES = 6 units?
A. 28 units
B. 24 units
C. 12 units
D. 9 units
14. A dart board has a diameter of 40 cm and is divided into 20 congruent
sectors. What is the area of one of the sectors?
A. 20 cm2
C. 80 cm2
B. 40 cm2
D. 800 cm2
15. Mr. Soriano wanted to plant three different colors of roses on the outer
rim of a circular garden. He stretched two strings from a point external
to the circle to see how the circular rim can be divided into three
portions as shown in the figure below.
What is the measure of minor arc AB?
A. 64° B. 104° C. 168° D. 192°
16. In the figure below, SY and EY are secants. If SY = 15 cm,
TY = 6 cm, and LY = 8 cm. What is the length of EY ?
A. 20 cm B. 12 cm C. 11.25 cm D. 6.75 cm
P
E R
O
S
N
S
Y
T
L
E
A
B
M
C
20°
192°

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17. In C below, mAB = 60 and its radius is 6 cm. What is the area of
the shaded region in terms of pi ( )?
A. 6  cm 2
C. 10  cm 2
B. 8  cm 2
D. 12  cm 2
18. In the circle below, what is the measure of SAY if DSY is a
semicircle and ?70SADm
A. 20 C. 110
B. 70 D. 150
19. Quadrilateral SMIL is inscribed in E. If 78m SMI and
m 95MSL  , find MILm .
A. 78 C. 95
B. 85 D. 102
20. In M on the right, what is BROm if 60BMOm ?
A. 120 C. 30
B. 60 D. 15
A
D
S
Y
C B
A
6 cm
60°
M
I
L
78°
S
95°
E
B
M
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R

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Part II
Solve each of the following problems. Show your complete solutions.
2. A bicycle chain fits tightly around two gears. What is the distance between the
centers of the gears if the radii of the bigger and smaller gears are 9.3 inches
and 2.4 inches, respectively, and the portion of the chain tangent to the two
gears is 26.5 inches long?
Rubric for Problem Solving
Score Descriptors
4
Used an appropriate strategy to come up with the correct solution
and arrived at a correct answer.
3
Used an appropriate strategy to come up with a solution, but a part
of the solution led to an incorrect answer.
2
Used an appropriate strategy but came up with an entirely wrong
solution that led to an incorrect answer.
1
Attempted to solve the problem but used an inappropriate strategy
that led to a wrong solution.
Part III
Read and understand the situation below, then answer the questions and
perform what is required.
The committee in-charge of the Search for the Cleanest and Greenest
School informed your principal that your school has been selected as a regional
finalist. Being a regional finalist, your principal would like to make your school
more beautiful and clean by making more gardens of different shapes. He
decided that every year level will be assigned to prepare a garden of particular
shape.
In your grade level, he said that you will be preparing circular,
semicircular, or arch-shaped gardens in front of your building. He further
encouraged your grade level to add garden accessories to make the gardens
more presentable and amusing.
1. Mr. Javier designed an arch made of
bent iron for the top of a school’s main
entrance. The 12 segments between
the two concentric semicircles are
each 0.8 meter long. Suppose the
diameter of the inner semicircle is 4
meters. What is the total length of the
bent iron used to make this arch?
4 m0.8 m

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1. How will you prepare the design of the gardens?
2. What garden accessories will you use?
3. Make the designs of the gardens which will be placed in front of your
grade level building. Use the different shapes that were required by your
principal.
4. Illustrate every part or portion of the garden including their measurements.
5. Using the designs of the gardens made, determine all the concepts or
principles related to circles.
6. Formulate problems involving these mathematics concepts or principles,
then solve.
Rubric for Design
Score Descriptors
4 The design is accurately made, presentable, and appropriate.
3 The design is accurately made and appropriate but not presentable.
2 The design is not accurately made but appropriate.
1 The design is made but not accurate and appropriate.
Rubric on Problems Formulated and Solved
Score Descriptors
6
Poses a more complex problem with two or more correct possible
solutions, communicates ideas unmistakably, shows in-depth
comprehension of the pertinent concepts and/or processes, and
provides explanations wherever appropriate.
5
Poses a more complex problem and finishes all significant parts of
the solution, communicates ideas unmistakably, and shows in-depth
comprehension of the pertinent concepts and/or processes.
4
Poses a complex problem and finishes all significant parts of the
solution, communicates ideas unmistakably, and shows in-depth
comprehension of the pertinent concepts and/or processes.
3
Poses a complex problem and finishes most significant parts of the
solution, communicates ideas unmistakably, and shows
comprehension of major concepts although neglects or misinterprets
less significant ideas or details.
2
Poses a problem and finishes some significant parts of the solution
and communicates ideas unmistakably but shows gaps on theoretical
comprehension.
1
Poses a problem but demonstrates minor comprehension, not being
able to develop an approach.
Source: D.O. #73, s. 2012
IV. LEARNING GOALS AND TARGETS
After going through this module, you should be able to demonstrate
understanding of key concepts of circles and formulate real-life problems
involving these concepts, and solve these using a variety of strategies.
Furthermore, you should be able to investigate mathematical relationships in
various situations involving circles.

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J
A N
E
L
s
Start Lesson 1A of this module by assessing your knowledge of the
different mathematical concepts previously studied and your skills in
performing mathematical operations. These knowledge and skills will
help you understand circles. As you go through this lesson, think of this
important question: “How do the relationships among chords, arcs, and
central angles of a circle facilitate finding solutions to real-life problems
and making decisions?” To find the answer, perform each activity. If you
find any difficulty in answering the exercises, seek the assistance of your
teacher or peers or refer to the modules you have studied earlier. You
may check your work with your teacher.
Use the figure below to identify and name the following terms related to A.
Then, answer the questions that follow.
1. a radius 5. a minor arc
2. a diameter 6. a major arc
3. a chord 7. 2 central angles
4. a semicircle 8. 2 inscribed angles
Activity 1:

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Questions:
a. How did you identify and name the radius, diameter, and chord?
How about the semicircle, minor arc, and major arc? inscribed angle
and central angle?
b. How do you describe a radius, diameter, and chord of a circle?
and central angle?
Write your answers in the table below.
Terms Related to Circles Description
1. radius
2. diameter
3. chord
4. semicircle
5. minor arc
6. major arc
7. central angle
8. inscribed angle
c. How do you differentiate among the radius, diameter, and chord of a
circle?
and central angle?
Were you able to identify and describe the terms related to circles?
Were you able to recall and differentiate them? Now that you know the
important terms related to circles, let us deepen your understanding of
finding the lengths of sides of right triangles. You need this mathematical
skill in finding the relationships among chords, arcs, and central angles
as you go through this lesson.

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In each triangle below, the length of one side is unknown. Determine the
length of this side.
1. 4.
2. 5.
3. 6.
Questions:
a. How did you find the missing side of each right triangle?
b. What mathematics concepts or principles did you apply to find each
missing side?
In the activity you have just done, were you able to find the missing
side of a right triangle? The concept used will help you as you go on with
this module.
Activity 2:
a = 6
b = 8
c = ?
b = ?
a = 3
c = 5
a = 9
b = 9
c = ?
a = ?
b = 16
c = 20
a = 7
b = ?
c = 14
a = 9
b = 15
c = ?

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O
Use the figures below to answer the questions that follow.
Figure 1 Figure 2
1. What is the measure of each of the following angles in Figure 1? Use a
protractor.
a. TOP d. ROS
b. POQ e. SOT
c. QOR
2. In Figure 2, AF , AB , AC , AD , and AE are radii of A. What is the
measure of each of the following angles? Use a protractor.
a. FAB d. EAD
b. BAC e. EAF
c. CAD
3. How do you describe the angles in each figure?
4. What is the sum of the measures of ,TOP POQ , ,QOR ,ROS
and SOT in Figure 1?
How about the sum of the measures of ,FAB ,BAC ,CAD ,EAD
and EAF in Figure 2?
Activity 3:
OT
S
R
Q
P
C
B
F
D
E
AO

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5. In Figure 1, what is the sum of the measures of the angles formed by
the coplanar rays with a common vertex but with no common interior
points?
6. In Figure 2, what is the sum of the measures of the angles formed by
the radii of a circle with no common interior points?
7. In Figure 2, what is the intercepted arc of ?FAB How about ?BAC
CAD ? ?EAD ?EAF Complete the table below.
Central Angle Measure Intercepted Arc
a. FAB
b. BAC
c. CAD
d. EAD
e. EAF
8. What do you think is the sum of the measures of the intercepted arcs
of ,FAB BAC , CAD , ,EAD and EAF ? Why?
9. What can you say about the sum of the measures of the central angles
and the sum of the measures of their corresponding intercepted arcs?
Were you able to measure the angles accurately and find the sum
of their measures? Were you able to determine the relationship between
the measures of the central angle and its intercepted arc? For sure you
were able to do it. In the next activity, you will find out how circles are
illustrated in real-life situations.
Use the situation below to answer the questions that follow.
Rowel is designing a mag wheel like the one shown below. He decided
to put 6 spokes which divide the rim into 6 equal parts.
Activity 4:

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In the figure on the right, BAC is a central
angle. Its sides divide A into arcs. One arc is the
curve containing points B and C. The other arc is the
curve containing points B, D, and C.
Recall that a central angle of a circle is an
angle formed by two rays whose vertex is the center
of the circle. Each ray intersects the circle at a point,
dividing it into arcs.
Questions:
a. What is the degree measure of each arc along the rim?
How about each angle formed by the spokes at the hub?
b. If you were to design a wheel, how many spokes will you use to divide
the rim? Why?
How did you find the preceding activities? Are you ready to learn
about the relations among chords, arcs, and central angles of a circle? I
am sure you are!!! From the activities done, you were able to recall and
describe the terms related to circles. You were able to find out how
circles are illustrated in real-life situations. But how do the relationships
among chords, arcs, and central angles of a circle facilitate finding
solutions to real-life problems and making decisions? You will find these
out in the activities in the next section. Before doing these activities, read
and understand first some important notes on this lesson and the
examples presented.
Central Angle and Arcs
Definition: Sum of Central Angles
(Note: All measures of angles and arcs are in degrees.)
The sum of the measures of the central angles
of a circle with no common interior points is 360
degrees.
In the figure, 3604m3m2m1m  .
4 3
21
B
A
C
D
central
angle
arc
arc

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M
O
G
E
Arcs of a Circle
An arc is a part of a circle. The symbol for arc is . A semicircle is
an arc with a measure equal to one-half the circumference of a circle. It is
named by using the two endpoints and another point on the arc.
Degree Measure of an Arc
1. The degree measure of a minor arc is the measure of the
central angle which intercepts the arc.
Example: GEO is a central angle. It intercepts E at
points G and O. The measure of GO is equal
to the measure of .GEO If m 118,GEO 
then mGO = 118.
2. The degree measure of a major arc is equal to 360 minus
the measure of the minor arc with the same endpoints.
Example: If mGO = 118, then mOMG = 360 – mGO.
That is, mOMG = 360 – 118 = 242.
Answer: mOMG = 242
3. The degree measure of a semicircle is 180 .
U
A minor arc is an arc of the circle that
measures less than a semicircle. It is named usually
by using the two endpoints of the arc.
Examples: JN, NE, and JE
A major arc is an arc of a circle that measures
greater than a semicircle. It is named by using the two
endpoints and another point on the arc.
Examples: JEN, JNE, and EJN
N
J
E
A
Z
O
C
N
Example: The curve from point N to point Z is an
arc. It is part of O and is named as
arc NZ or NZ. Other arcs of O are
CN, CZ, CZN, CNZ, and NCZ.
If mCNZ is one-half the circumference
of O, then it is a semicircle.

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W
T
S
E
65°
65°
N
M
K
I
M
A
T
H
Congruent Circles and Congruent Arcs
Congruent circles are circles with congruent radii.
Example: MA is a radius of A.
TH is a radius of T.
If THMA  , then A T.
Congruent arcs are arcs of the same circle or of
congruent circles with equal measures.
Example: In I, KSTM  .
If I E, then NWTM 
and NWKS  .
Theorems on Central Angles, Arcs, and Chords
1. In a circle or in congruent circles, two minor arcs are congruent if and only
if their corresponding central angles are congruent.
In E below, NEOSET  Since the two central angles are
congruent, the minor arcs they intercept are also congruent. Hence,
NOST  .
If E  I and BIGNEOSET  , then BGNOST  .
65°
T
50°50°
T
S N
O
E I
G
B
50°
.

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Proof of the Theorem
The proof has two parts. Part 1. Given are two congruent circles and a
central angle from each circle which are congruent. The two-column proof
below shows that their corresponding intercepted arcs are congruent.
Given: E I
BIGSET 
Prove: BGST 
Proof:
Statements Reasons
1. E I
BIGSET 
1. Given
2. In E , .m SET mST 
In I , .m BIG mBG 
2. The degree measure of a minor arc is
the measure of the central angle
which intercepts the arc.
3. BIGmSETm  3. From 1, definition of congruent angles
4. mBGmST  4. From 2 & 3, substitution
5. BGST  5. From 4, definition of congruent arcs
Part 2. Given are two congruent circles and intercepted arcs from each
circle which are congruent. The two-column proof on the next page shows
that their corresponding angles are congruent.
Given: E I
BGST 
Prove: BIGSET 
G
B
I
ES
T
G
B
I
ES
T

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B
A
H
E
O
N
C
T
Proof:
Statements Reasons
1. E I
BGST 
1. Given
2. In ,E .mST m SET 
In I , .mBG m BIG 
3. mBGmST  3. From 1, definition of congruent arcs
4. BIGmSETm  4. From 2 & 3, substitution
5. BIGSET  5. From 4, definition of congruent angles
Combining parts 1 and 2, the theorem is proven.
2. In a circle or in congruent circles, two minor arcs are
congruent if and only if their corresponding chords are
congruent.
In T on the right, CHBA  . Since
the two chords are congruent, then CHBA  .
If T  N and OECHBA  , then
OECHBA  .

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B
A
T
E
N
O
B
A
T E
N
O
Proof of the Theorem
The proof has two parts. Part 1. Given two congruent circles
T N and two congruent corresponding chords AB and OE , the two-
column proof below shows that the corresponding minor arcs AB and OE are
congruent.
Given: T N
OEAB 
Prove: OEAB 
Proof:
Statements Reasons
1. T N
OEAB 
1. Given
2. NENOTBTA 
2. Radii of the same circle or of
congruent circles are congruent.
3. ONEATB  3. SSS Postulate
4. ONEATB 
4. Corresponding Parts of Congruent
Triangles are Congruent (CPCTC)
5. OEAB 
5. From the previous theorem, “In a
circle or in congruent circles, two
minor arcs are congruent if and only
if their corresponding central angles
are congruent.”
Part 2. Given two congruent circles T and N and two congruent minor
arcs AB and OE , the two-column proof on the next page shows that the
corresponding chords AB and OE are congruent.
Given: T N
OEAB 
Prove: OEAB 

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Proof:
Statements Reasons
1. T N
OEAB 
1. Given
2. mOEmAB  2. Definition of congruent arcs
3. BTA and ONE are
central angles.
3. Definition of central angles
4. mBABTAm 
mOEONEm 
5. ONEmBTAm  5. From 2, 4, substitution
6. NENOTBTA 
6. Radii of the same circle or of
congruent circles are congruent.
7. ONEATB  7. SAS Postulate
8. OEAB 
8. Corresponding Parts of Congruent
Triangles are Congruent (CPCTC)
Combining parts 1 and 2, the theorem is proven.
3. In a circle, a diameter bisects a chord and an arc with the same endpoints
if and only if it is perpendicular to the chord.
The proof of the theorem is given as an exercise in Activity 9.
N
E
G
U
I
S
In U on the right, ES is a diameter and
GN is a chord. If GNES  , then INGI  and
ENGE  .

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Arc Addition Postulate
The measure of an arc formed by two adjacent arcs
is the sum of the measures of the two arcs.
Example: Adjacent arcs are arcs with exactly one
point in common. In E, LO and OV
are adjacent arcs. The sum of their
measures is equal to the measure of
LOV.
If mLO = 71 and mOV = 84, then mLOV = 71 + 84 = 155.
Sector and Segment of a Circle
A sector of a circle is the region bounded by an arc of the circle and
the two radii to the endpoints of the arc. To find the area of a sector of a
circle, get the product of the ratio
360
arctheofmeasure
and the area of the
circle.
Example: The radius of C is 10 cm. If mAB = 60, what is the
area of sector ACB?
Solution: To find the area of sector ACB:
a. Determine first the ratio
360
arctheofmeasure
.
360
60
360

arctheofmeasure
6
1

b. Find the area (A) of the circle using the equation A =
2
r ,
where r is the length of the radius.
A =
2
r
=  2
cm10
= 2
cm100
O
E
V
L
C B
A
10 cm
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c. Get the product of the ratio
360
arctheofmeasure
and the
area of the circle.
Area of sector ACB =  2
cm100
6
1






= 2
cm
3
50
The area of sector ACB is 2
cm
3
50
.
A segment of a circle is the region bounded by an arc and the
segment joining its endpoints.
Example: The shaded region in the figure below is a segment of
T. It is the region bounded by PQ and PQ .
In the same figure, the area of ΔPTQ =   cm5cm5
2
1
or
ΔPTQ = 2
cm
2
25
.
The area of the shaded segment, then, is equal to 2
cm
2
25
4
25

which is approximately 7.135 cm2
.
To find the area of the shaded segment
in the figure, subtract the area of triangle PTQ
from the area of sector PTQ.
If mPQ = 90 and the radius of the circle
is 5 cm, then the area of sector PTQ is one-
fourth of the area of the whole circle. That is,
Area of sector PTQ =   




 2
5
4
1
cm
=  




 2
cm25
4
1
= 2
cm
4
25


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Arc Length
The length of an arc can be determined by using the proportion
360 2


A
r
l
, where A is the degree measure of the arc, r is the radius of the
circle, and l is the arc length. In the given proportion, 360 is the degree
measure of the whole circle, while 2r is the circumference.
Example: An arc of a circle measures 45°. If the radius of the circle is 6 cm,
what is the length of the arc?
Solution: In the given problem, A = 45 and r = 6 cm. To find l, the equation
360 2


A
r
l
can be used. Substitute the given values in the
equation.
360 2


A
r
l

45
360 2 (6)


l

1
8 12


l
12
8

 l  4.71l
The length of the arc is approximately 4.71 cm.
Learn more about Chords,
Arcs, Central Angles,
Sector, and Segment of a
Circle through the WEB.
You may open the
following links.
http://www.cliffsnotes.com/math/geometry/
circles/central-angles-and-arcs
http://www.mathopenref.com/arc. html
http://www.mathopenref.com/chord.html
http://www.mathopenref.com/circlecentral. html
http://www.mathopenref.com/arclength.html
http://www.mathopenref.com/arcsector.html
http://www.mathopenref.com/segment.html
http://www.math-worksheet.org/arc-length-and-
sector-area

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Your goal in this section is to apply the key concepts of chords,
arcs, and central angles of a circle. Use the mathematical ideas and the
examples presented in the preceding section to answer the activities
provided.
Use A below to identify and name the following. Then, answer the questions
that follow.
1. 2 semicircles in the figure
2. 4 minor arcs and their corresponding
major arcs
3. 4 central angles
Questions:
a. How did you identify and name the semicircles?
How about the minor arcs and the major arcs? central angles?
b. Do you think the circle has more semicircles, arcs, and central
angles? Show.
Were you able to identify and name the arcs and central angles in
the given circle? In the next activity, you will apply the theorems on arcs
and central angles that you have learned.
Activity 5:
H
K L
M
G
A
J

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In A below, m 42,LAM m 30,HAG and KAH is a right angle. Find the
following measure of an angle or an arc, and explain how you arrived at your
answer.
1. LAKm 6. LKm
2. JAKm 7. JKm
3. LAJm 8. LMGm
4. JAHm 9. JHm
5. KAMm 10. KLMm
In the activity you have just done, were you able to find the degree
measure of the central angles and arcs? I am sure you did! In the next
activity, you will apply the relationship among the chords, arcs, and
central angles of a circle.
In the figure, JI and ON are diameters of S. Use the figure and the given
information to answer the following.
1. Which central angles are congruent? Why?
2. If 113m JSN , find:
a. ISOm
b. NSIm
c. JSOm
3. Is INOJ  ? How about JN and OI ? Justify your answer.
4. Which minor arcs are congruent? Explain your answer.
5. If 67m JSO , find:
a. JOm d. IOm
b. JNm e. JOmN
c. NIm f. NIOm
6. Which arcs are semicircles? Why?
Activity 7:
Activity 6:
I
J
S
N
O
H
K L
M
G
A
J

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Were you able to apply the relationship among the chords, arcs and
central angles of a circle? In Activity 8, you will use the theorems on
chords in finding the lengths of chords.
In M below, BD = 3, KM = 6, and KP = 72 . Use the figure and the given
information to find each measure. Explain how you arrived at your answer.
1. AM 5. DS
2. KL 6. MP
3. MD 7. AK
4. CD 8. KP
Were you able to find the length of the segments? In the next
activity, you will complete the proof of a theorem on central angles, arcs,
and chords of a circle.
Complete the proof of the following theorem.
In a circle, a diameter bisects a chord and an arc with the same
endpoints if and only if it is perpendicular to the chord.
Given: ES is a diameter of U and
perpendicular to chord GN at I.
Prove: 1. GINI 
2. EGEN 
3. GSNS 
Activity 9:
Activity 8:
D
M
3
C
6
72
A
S
P
L
B
K
N
G
E
I
S
U

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Proof of Part 1: Show that ES bisects GN and the minor arc GN.
Statements Reasons
1. U with diameter ES and chord
GN GNES 
Two points determine a line.
2. GIU and NIU are right angles. Given
3. Lines that are perpendicular tm
right angles.
4. UNUG  Radii of a circle are congruent.
5. UIUI  Reflexive Property of Congruence.
6. NIUGIU 
7. NIGI 
Corresponding parts of congruent
triangles are congruent.
8. ES bisects GN .
triangles are congruent
9. NUIGUI 
In a circle, congruent central angles
intercept congruent arcs.
10. GUI and GUE are the same
angles.
NUI and NUE are the same
angles.
Two angles that form a linear pair
are supplementary.
11. NUEmGUEm 
Supplements of congruent angles
are congruent.
12. GUEmmEG 
NUEmmEN 
In a circle, congruent central angles
intercept congruent arcs.
13. mEGmEN 
14. NUSmGUSm 
15. GUSmmGS 
NUSmmNS 
16. mGSmNS 
17. ES bisects GN .
;
GIU NIU  

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Given: ES is a diameter of U; ES bisects GN
at I and the minor arc GN.
Proof of Part 2: Show that GNES  .
Statements Reasons
1. U with diameter ES , ES bisects
GN at I and the minor arc
GN.
Two points determine a line.
2. NIGI 
NEGE 
Given
3.
4. UNUG  Radii of a circle are congruent.
5. NIUGIU  Reflexive Property of Congruence.
6. UINUIG 
7. UIG and UIN are right
angles.
triangles are congruent.
8. GNIU 
9. GNES 
Was the activity interesting? Were you able to complete the proof?
You will do more of this in the succeeding lessons. Now, use the ideas
you have learned in this lesson to find the arc length of a circle.
S
N
GI
U
E
UI UI
S

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The radius of O below is 5 units. Find the length of each of the following
arcs given the degree measure. Answer the questions that follow.
1. mPV = 45; length of PV = ________
2. mPQ = 60; length of PQ = ________
3. mQR = 90; length of QR = ________
4. mRTS = 120;length of RTS = ________
5. mQRT = 95; length of QRT = ________
Questions:
a. How did you find the length of each arc?
b. What mathematics concepts or principles did you apply to find the
length of each arc?
Were you able to find the arc length of each circle? Now, find the
area of the shaded region of each circle. Use the knowledge learned
about segment and sector of a circle in finding each area.
Find the area of the shaded region of each circle. Answer the questions that
follow.
1. 2. 3.
Activity 11:
Activity 10:
BC
A
R
Q
S
X
Z Y6 cm
90°
12 cm
45°
8 cm
135°
T
S
P
Q
R
Or = 5
V

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4. 5. 6.
Questions:
a. How did you find the area of each shaded region?
b. What mathematics concepts or principles did you apply to find the area
of the shaded region? Explain how you applied these concepts.
How was the activity you have just done? Was it easy for you to find
the area of segments and sectors of circles? It was easy for sure!
In this section, the discussion was about the relationship among
chords, arcs, and central angles of circles, arc length, segment and
sector of a circle, and the application of these concepts in solving
problems.
Go back to the previous section and compare your initial ideas with
the discussion. How much of your initial ideas are found in the
discussion?
Now that you know the important ideas about this topic, let us go
deeper by moving on to the next section.
T
X
S
4 cm
A
R
E
B
W M5 cm
100°
J
S
O
6 cm
Y

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Your goal in this section is to take a closer look at some aspects of the
topic. You are going to think deeper and test further your understanding of
circles. After doing the following activities, you should be able to answer this
important question: “How do the relationships among chords, arcs, and
central angles of a circle facilitate finding solutions to real-life problems and
making decisions?”
Answer the following questions.
1. Five points on a circle separate the circle into five congruent arcs.
a. What is the degree measure of each arc?
b. If the radius of the circle is 3 cm, what is the length of each arc?
c. Suppose the points are connected consecutively with line segments. How
do you describe the figure formed?
2. Do you agree that if two lines intersect at the center of a circle, then the lines
intercept two pairs of congruent arcs? Explain your answer.
3. In the two concentric circles on the right,
CON intercepts CN and RW.
a. Are the degree measures of CN and RW
equal? Why?
b. Are the lengths of the two arcs equal?
Explain your answer.
4. The length of an arc of a circle is 6.28 cm. If the circumference of the circle is
37.68 cm, what is the degree measure of the arc? Explain how you arrived at
your answer.
5.
Activity 12:
O
R
W N
C
Mr. Lopez would like to place a fountain in his
circular garden on the right. He wants the pipe,
where the water will pass through, to be located
at the center of the garden. Mr. Lopez does not
know where it is. Suppose you were asked by
Mr. Lopez to find the center of the garden, how
would you do it?

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6. The monthly income of the Soriano family is Php36,000.00. They spend
Php9,000.00 for food, Php12,000.00 for education, Php4,500.00 for
utilities, and Php6,000.00 for other expenses. The remaining amount is for
their savings. This information is shown in the circle graph below.
a. Which item is allotted with the highest budget? How about the least?
Explain.
b. If you were to budget your family’s monthly income, which item would
you give the greater allocation? Why?
c. In the circle graph, what is the measure of the central angle
corresponding to each item?
d. How is the measure of the central angle corresponding to each item
determined?
e. Suppose the radius of the circle graph is 25 cm. What is the area of
each sector in the circle graph? How about the length of the arc of
each sector?
In this section, the discussion was about your understanding of
chords, arcs, central angles, area of a segment and a sector, and arc
length of a circle including their real-life applications.
What new realizations do you have about the lesson? How would
you connect this to real life?
Now that you have a deeper understanding of the topic, you are
ready to do the tasks in the next section.
Soriano Family’s
Monthly Expenses

Math lm unit 2

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