(8) Inquiry Lab - Proofs About the Pythagorean Theorem

1. HOW can you prove the
Pythagorean Theorem
and its converse?
Course 3, Inquiry Lab after Lesson 5-5
Geometry

2. • 8.G.6
Explain a proof of the Pythagorean Theorem and its converse.
Mathematical Practices
1 Make sense of problems and persevere in solving them.
3 Construct viable arguments and critique the reasoning of others.
7 Look for and make use of structure.
Course 3, Inquiry Lab after Lesson 5-5 Common Core State Standards © Copyright 2010. National Governors Association Center for
Best Practices and Council of Chief State School Officers. All rights reserved.
Geometry

3. Course 3, Inquiry Lab after Lesson 5-5
Activity 1
continued

Geometry
The Pythagorean Theorem is named after the famous Greek
mathematician Pythagoras who lived around 500 B.C. The properties of the
theorem, however, were known by the ancient Egyptians, Babylonians,
and Chinese. The following geometric proof is similar to a visual proof
shown in a Chinese document written between 500 B.C. and 200 B.C.
Draw and cut out 8 copies of a right triangle. Label leach pair of
legs a and b, and each hypotenuse c.

4. Course 3, Inquiry Lab after Lesson 5-5
Activity1
continued

Geometry
On a separate piece of paper arrange four of the
triangles in a square as shown. Trace the figure
formed by the hypotenuses.
The length of each side of the large square is a + b,
so the area of the large square is 2
( ) .a b
Is the figure formed by the hypotenuses a square? Explain.
_________________________________________________________
_________________________________________________________
_________________________________________________________
Write an expression for the area of the inside square. _______________

5. Course 3, Inquiry Lab after Lesson 5-5
Activity 1
continued

Geometry
On the same paper, arrange the remaining
triangles as shown. Draw the two figures
shown by the dashed lines.
The length of each side of the large square is a + b,
so the area of the large square is 2
( ) .a b
Are the two figures represented by dashed lines squares? Explain.
_______________________________________________________
_______________________________________________________
Write an expression for the area of the small square.
Write an expression for the area of the large square.

6. Course 3, Inquiry Lab after Lesson 5-5
Geometry
Activity 1
continued

Since the area of each of the two composite figures you
created is , the areas are equal. Use the space
provided to draw each figure from Step 2 and Step 3. Place
an equal sign between the two figures to show the two areas
are equal.
2
( )a b

7. Course 3, Inquiry Lab after Lesson 5-5
Geometry
Remove the triangles from each side. Use the space provided
to draw the remaining figures.
What property justifies removing the triangles from each side of the
equation? ___________________________________________________
Write an algebraic equation that represent the relationship between the
figures shown in Step 5. ________________________________________
Summarize the relationship among the sides of a right triangle measuring
a units, b units, and c units. _____________________________________
___________________________________________________________

8. Course 3, Inquiry Lab after Lesson 5-5
Activity 2
continued

Geometry
The converse of the Pythagorean Theorem states if a triangle has
side lengths, a, b, and c such that , then the triangle is a
right triangle. In this Activity, you will prove the converse of the
Pythagorean Theorem by using a two-column proof.
2 2 2
a b c 
Given: .
Prove: is a right triangle.
such thatABC 2 2 2
a b c 
ABC
Complete the proof with the correct reasons justifying each statement.
_____________________________
Statements Reasons
a. Draw a right triangle DEF
so that is b units long.
Label as d.
DF
FE

9. Course 3, Inquiry Lab after Lesson 5-5
Activity 2
continued

Geometry
Statements Reasons
The converse of the Pythagorean Theorem states if a triangle has side
lengths, a, b, and c such that , then the triangle is a right triangle.
In this Activity, you will prove the converse of the Pythagorean Theorem by
using a two-column proof.
Given: .
Prove: is a right triangle.
Complete the proof with the correct reasons justifying each statement.
such thatABC 2 2 2
a b c 
ABC
2 2 2
a b c 
b. Write an equation that
describes the relationship
between the side lengths of
. State the theorem
that allows you to make that
statement.
DEF
______________________
______________________
______________________

10. Course 3, Inquiry Lab after Lesson 5-5
Activity 2
continued

Geometry
The converse of the Pythagorean Theorem states if a triangle has side
lengths, a, b, and c such that , then the triangle is a right triangle.
In this Activity, you will prove the converse of the Pythagorean Theorem by
using a two-column proof.
Given: .
Prove: is a right triangle.
Complete the proof with the correct reasons justifying each statement.
2 2 2
a b c 
such thatABC 2 2 2
a b c 
ABC
Statements Reasons
2 2 2
a b c  Givenc.

11. Course 3, Inquiry Lab after Lesson 5-5
Activity 2
continued

Geometry
The converse of the Pythagorean Theorem states if a triangle has side
lengths, a, b, and c such that , then the triangle is a right triangle.
In this Activity, you will prove the converse of the Pythagorean Theorem by
using a two-column proof.
Given: .
Prove: is a right triangle.
Complete the proof with the correct reasons justifying each statement.
2 2 2
a b c 
such thatABC 2 2 2
a b c 
ABC
Statements Reasons
d. If and
then .
2 2 2
a b c 
2 2 2
a b d  2 2
d c
______________________
______________________
______________________

12. Course 3, Inquiry Lab after Lesson 5-5
Activity 2
continued

Geometry
The converse of the Pythagorean Theorem states if a triangle has side
lengths, a, b, and c such that , then the triangle is a right triangle.
In this Activity, you will prove the converse of the Pythagorean Theorem by
using a two-column proof.
Given: .
Prove: is a right triangle.
Complete the proof with the correct reasons justifying each statement.
Statements Reasons
2 2 2
a b c 
such thatABC 2 2 2
a b c 
ABC
e. If , then d = c.2 2
d c ______________________
______________________
______________________

13. Course 3, Inquiry Lab after Lesson 5-5
Activity 2
continued

Geometry
The converse of the Pythagorean Theorem states if a triangle has side
lengths, a, b, and c such that , then the triangle is a right triangle.
In this Activity, you will prove the converse of the Pythagorean Theorem by
using a two-column proof.
Given: .
Prove: is a right triangle.
Complete the proof with the correct reasons justifying each statement.
Statements Reasons
______________________
______________________
______________________
2 2 2
a b c 
such thatABC 2 2 2
a b c 
ABC
f. If d = c then FE = AB.

14. Course 3, Inquiry Lab after Lesson 5-5
Activity 2
continued

Geometry
The converse of the Pythagorean Theorem states if a triangle has side
lengths, a, b, and c such that , then the triangle is a right triangle.
In this Activity, you will prove the converse of the Pythagorean Theorem by
using a two-column proof.
Given: .
Prove: is a right triangle.
Complete the proof with the correct reasons justifying each statement.
Statements Reasons
2 2 2
a b c 
such thatABC 2 2 2
a b c 
ABC
g. If AC = FD, CB = DE, and
AB = FE, the two triangles
are the same shape and
size.
If three sides of a triangle are
the same length as the
corresponding sides of
another triangle, the triangles
are the same shape and size.

15. Course 3, Inquiry Lab after Lesson 5-5
Activity 2
continued

Geometry
The converse of the Pythagorean Theorem states if a triangle has side
lengths, a, b, and c such that , then the triangle is a right triangle.
In this Activity, you will prove the converse of the Pythagorean Theorem by
using a two-column proof.
Given: .
Prove: is a right triangle.
Complete the proof with the correct reasons justifying each statement.
Statements Reasons
Corresponding parts of the
triangles with the same size and
shape have the same measure.
2 2 2
a b c 
such thatABC 2 2 2
a b c 
ABC
h. m C m D  

16. Course 3, Inquiry Lab after Lesson 5-5
Activity 2
continued

Geometry
The converse of the Pythagorean Theorem states if a triangle has side
lengths, a, b, and c such that , then the triangle is a right triangle.
In this Activity, you will prove the converse of the Pythagorean Theorem by
using a two-column proof.
Given: .
Prove: is a right triangle.
Complete the proof with the correct reasons justifying each statement.
Statements Reasons
2 2 2
a b c 
such thatABC 2 2 2
a b c 
ABC
i. is a right angle.C ______________________
______________________
______________________

17. Course 3, Inquiry Lab after Lesson 5-5
Geometry
The converse of the Pythagorean Theorem states if a triangle has side
lengths, a, b, and c such that , then the triangle is a right triangle.
In this Activity, you will prove the converse of the Pythagorean Theorem by
using a two-column proof.
Given: .
Prove: is a right triangle.
Complete the proof with the correct reasons justifying each statement.
Statements Reasons
______________________
______________________
2 2 2
a b c 
such thatABC 2 2 2
a b c 
ABC
j. is a right triangle.ABC
So, if a triangle has side lengths, a, b, and c units such that ,
then the triangle is a right triangle.
2 2 2
a b c 

18. Course 3, Inquiry Lab after Lesson 5-5
HOW can you prove the
Pythagorean Theorem
and its converse?
Geometry