1. Circles and some related terms
A circle is defined as a
set of all such points in a
given plane which lie at
a fixed distance from a
fixed point in the plane.
This fixed point is called
the center of the circle
and the fixed distance is
called the radius of the
circle
where point P is the center of the circle and segment PQ is known as the
radius. The radius is the distance between all points on the circle and P. It
follows that if a R exists such that (seg.PQ) > (seg.PR) the R is inside the
circle. On the other hand for a T if (seg.PT) > (seg.PQ) T lies outside the
circle, and if (seg.PS) = (seg.PQ) it can be said that S lies on the circle.
2. Lines of a Circle
The lines in the plane of the
circle are classified into
three categories.
a) Lines like l which do not
intersect the circle.
b) Lines like m which
intersect the circle at only
one point.
c) Lines like n which
intersect the circle at two
points..
3. Lines of a Circle
Lines like m are
called tangents. A
tangent is a line that
has one of its points
on a circle and the
rest outside the
circle. Thus K is the
point of tangency.
Line n is called a secant of the circle. A
secant is defined as any line that intersects a
circle in two distinct points.
K
4. Lines of a Circle
A segment whose
end points lie on a
circle is called
a Chord . In a figure
AB is a chord of the
circle. Thus a chord is
always a part of
secant.
Thus the other chords are;
5. Lines of a Circle
The longest chord of the
circle passes through its
center and is called as
the diameter. In the
figure chord CD is the
diameter. It can be
noticed immediately that
the diameter is twice the
radius of the circle. The
center of the circle is the
midpoint of the diameter.
6. Example 1
Refer to ⊙ 𝑂.
1. Name the center of ⊙
2. Name the longest
chord
3. Name three radii
4. Name a secant
5. Name a tangent and
the point of tangency.
6. If OC=12, find SI.
7. Is OS a chord of ⊙ 𝑂.
8. Is SI>ER? Explain.
7. Arcs
The angle described by
any two radii of a circle
is called the central
angle. Its vertex is the
center of the circle. ∠APB
is a central angle. The
part of the circle that is
cut by the arms of the
central angle is called an
arc. AB is an arc
And the other arcs are;
8. Arcs
arcAB is called the
minor arc and is the
arcAOB is a major arc.
The minor arc is always
represented by using the
two end points of the arc
on the circle. However it
is customary to denote
the major arc using three
points. The two end
points of the major arc
and a third point also on
the arc.
9. Arcs
If a circle is cut into two
arcs such that there is no
minor or major arc but
both the arcs are equal
then each arc is called
a semicircle.
10. Arcs
An arc is measured as an
angle in degrees and also
in units of length. The
measure of the angle of
an arc is its central angle
and the length of the arc
is the length of the
portion of the
circumference that it
describes.
If ∠𝐴𝑃𝐵 is a central angle, then 𝑚∠𝐴𝑃𝐵 = 𝐴𝐵
11. The Arc Addition Postulate
Given point B on 𝐴𝐶,
then 𝑚 𝐴𝐶=mAB + mBC.
12. Example 2
Identify the following.
1. 2 major arcs
2. 2 minor arcs
3. An acute central angle
4. An obtuse central
angle
5. A radius which is not
a part of a diameter
6. A semicircle
13. Inscribed angles
An inscribed angles are
formed by chords. the
vertex O of the inscribed
∠ AOB is on the circle.
The minor arc AB cut on
the circle by an inscribed
angle is called as the
intercepted arc.
14. The Inscribed Angle Theorem
The measure of an
inscribed angle is half
the measure of its
intercepted arc.
If ∠𝐴𝑂𝐵 is an inscribed angle, then 𝑚∠𝐴𝑂𝐵 =
1
2
𝐴𝐵
15. Example 3
In the figure at the right,
𝑚∠𝐴𝑇𝑀=72. Find
a. 𝑚 𝐴𝑀
b. 𝑚∠𝑈
c. 𝑚∠𝑂
16. Inscribed Angle
Theorem:
If two inscribed angles
intercept the same arc or
arcs of equal measure
then the inscribed angles
have equal measure.
If ∠𝐶𝐴𝐷 and ∠𝐶𝐵𝐷 are inscribed angles with same
intercepted arc CD, then ∠𝐶𝐴𝐷 ≅ ∠𝐶𝐵𝐷.
