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Summary
mary
+ mary

Prepared by: MIKEE C. TOLENTINO

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 identify and illustrate parts of a
circle; and
 apply theorems in solving
problems in tangent, secant
segments, arcs, chords and angles
of a circle.

CIRCLE
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It is the set of all
points on a plane at a
given distance from a
fixed point called the
center.

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 A circle is named by
its center.
Example: ʘA A
center

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RADIUS
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It is any segment
joining the center to a
point on
the circle.
radius
A B
Example: AB

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Interior of a circle
It is the set of all points in the
plane of the circle whose distance
from the center is
less than the radius. A
C
Example: C

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Exterior of a circle
It is the set of all points outside
the plane of the circle whose
distance from the center
is greater than the radius. A
D
Example: D

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Congruent Circles
It composed of two or more
circles having congruent radii.
Example: ʘA ≅ʘZ
A
Z

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Concentric Circles
It composed of two or more
coplanar circles having the same
center.

CHORD
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It is a segment
joining any two
points on
the circle. chord
E F
Example: EF

DIAMETER
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It is a chord passing
through the center. It
twice the
measure of
the radius.
diameter
H
G
Example: GH

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1. Name of the circle
2. Three radii
3. Length of radius
4. A diameter
5. A chord
6. Length of QS
7. Length of QX
Let’s Try It:
Q S X
T
R
P

ARC
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It is a part of a
circle between two
points on the circle.
I
J
Example: IJ

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Semicircle
It is one half of a circle and
measures 180⁰.
A diameter divides the
circle into two semicircles.
K
L
M
N
Example: NKL and LMN

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Major Arc
It is an arc of a circle
having a measure greater
than 180°.
Example: OPQ
O
P Q

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Minor Arc
It is an arc smaller than a
semicircle. It measures less
than 180°.
Example: QO
O
P Q

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Arc Addition Postulate
The measure of an arc
formed by two adjacent non-
overlapping arcs is the sum
of the measures of those arcs.

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Example 1:
If VR = 54⁰;RS = 43⁰
What is VRS?
Solution:
mVR+mRS = mVRS
54⁰ + 43⁰ = 97⁰
R
S
U
V

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Example 2:
If RSU =176⁰;RS = 49⁰
What is mSU? R
S
U
V

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Intercepted Arc
The arc that lies in the
interior of an angle and has
endpoints on
the angle.
Y
intercepted arc

ANGLE
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Central Angle
An angle whose vertex
is the center of the circle
and whose sides are radii
of the circle.

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Example:
 Central angle and its
intercepted arc are same in
measure. Thus, m WAX = mWX
central angle
intercepted arc A
W
X

Theorems on Arcs,
Chords, and
Central Angles
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Theorem 1: In the same circle, or
congruent circles, two minor arcs are
congruent if and only if their central
angles are congruent.
If, XAY≅ YAZ
Then, XY≅ YZ 31
A
X
Y
Z

Theorem 2: In the same circle
or congruent circles, two minor
arcs are congruent if and only if
their chords are
congruent.
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A
X
Y
Z

Theorem 3: If a diameter is
perpendicular to a chord, then it
bisects the chord and
its arc.
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A
X
Y
Z
W

Theorem 4: In the same circle
or congruent circles, two chords are
equidistant from the center(s) if and
only if they are
congruent.
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A
X
Y
Z
W
U
V

Theorem 5: If two chords of a
circle are unequal in length, then
the longer chord is nearer to the to
the center of the
circle.
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A
X
Y
Z
W
U
V

Theorem 6: If two chords of a
circle are not equidistant from the
center, then the longer chord is
nearer to the center
of the circle.
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A
X
Y
Z
W
U
V

ANGLE
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Inscribed Angle
An angle whose vertex is
on a circle and whose
sides contain chords of
the circle.

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Inscribed Angle Theorem
The measure of an angle is half the
measure of its intercepted arc.
inscribed angle
intercepted arc
Y
W
X

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Solution:
½ WX= WXY
2 (½ WX= 45⁰ ) 2
WX = 90⁰
Y
W
X
45⁰
Example:
What is WX?

Corollary 1:
If two inscribed
angles of a circle
intercept the same
arc or congruent
arcs, then the angles
are congruent. 40
C
D
B
A
E

C
D
B
A
Corollary 2:
If a quadrilateral is
inscribed in a circle ,
then its opposite
angles are
supplementary.
41
E

Corollary 3:
If an inscribed angle
intercepts a
semi-circle, then
the angle is a
right angle.
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C
D
B
A E

Corollary 4:
If two arcs of a
circle are included
between parallel
segments, then the
arcs are congruent.
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E
C
D
B
A

SECANT
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It is any line, ray, or
segment that intersects
circle in two
points.
J
I
Example: IJ

TANGENT
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It is a line that
intersects at exactly one
point on a circle. The
point is called the point
of tangency.

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Example:
BC is a tangent
BC is a tangent ray
C is the point
of tangency
BC is a tangent segment
B
C

Theorem 1: If a line is tangent to a circle,
then the line is perpendicular to the radius at the
point of tangency.
Theorem 2: If a line in the plane of a
circle is perpendicular to a radius at
its endpoint on the circle, then
the line is the tangent to the circle.
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B
C

Corollary 1: Two tangent
segments from a common external
point are congruent.
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A
C
B
D

Corollary 2: The two tangent rays from
a common external point determine an
angle that is bisected by the ray from the
external point to the center of the circle.
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A
C
B
D

Theorems on
Tangent, Secants
and Angle
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Theorem 1: If two chords intersect
within a circle, then the measure of the
angle formed is equal to one-half the sum
of the measures of the intercepted arcs.
If BE = 38⁰; CD = 76⁰
CAD = ½ (38⁰ +76⁰)
= 57⁰
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B A
C
D
E

Theorem 2: If a tangent and a chord
intersect in a point on the circle, then the
measure of the angle they form is one-half the
measure of the intercepted arc.
If BA = 84⁰
BCA = ½ (84⁰)
= 42⁰ 52
B
A
C
D

Theorem 3: If a tangent and a secant, two
secants , or two tangents intersect in a point in
the exterior of a circle, then the measure of the
angle formed is equal to one-half the difference
of the measures of the intercepted arcs.
If BE = 70⁰; BCA=106⁰
BDA = ½ (106⁰ - 70⁰)
= 18⁰
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B
A
C
D
E

Theorems on
Circles and
Segment Lengths
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Theorem 1: If two chords
intersect inside a circle, then the
product of the lengths of the
segments of one chords is equal to
the product of the lengths of the
segments of the other chord. 55

Example: Find the value of AE.
If BA= 4, AC = 6 and DA = 3.
BA(AC)= DA(AE)
(4)(6) = (3)(AE)
24 = 3AE
8 = AE 56
B
A
C
D
E

Theorem 2: If a secant and a
tangent intersect in the exterior of a
circle, then the product of the lengths
of the secant segment and its external
segment is equal to the square of the
length of the tangent segment.
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Example: Find the value of
DA. If BA= 9, CA = 16.
(𝐷𝐴)2
= CA(BA)
(𝐷𝐴)2
= (16) (9)
(𝐷𝐴)2
= 144
DA = 12 58
B
A
C
D

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1. Radii of the same circle are equal in measure.
2. Every chord is a diameter.
3. Central angle is twice the measure of its
intercepted arc.
4. Major arc measures greater than 180 degree but
less than 360 degree.
5. Inscribed angle is one-half the measure of its
intercepted arc.
Seatwork:

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6. Diameter is thrice the measure of a
radii.
7. Minor arc measures less than 90
degree.
8. A tangent passes through the center
of a circle.
Seatwork:

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Seatwork:
C
x
B
A
80⁰
D
9.
E

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Seatwork:
C
x A
65⁰
D
10. 11.
x
16⁰
78⁰

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Seatwork:
C
53⁰
D
12. 13. x
4
5
x
99⁰
62⁰

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Seatwork:
C
D
14. 15. x 6
12
x
8

“
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