1. The document defines key terms related to circles such as radius, diameter, chord, arc, tangent, secant, and angles related to circles. 2. Theorems regarding relationships between arcs, chords, angles and segment lengths in circles are presented along with examples of applying the theorems. 3. Exercises are provided for students to work through applying the concepts and theorems to problem solving.

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Summary
mary
+ mary

7. Prepared by: MIKEE C. TOLENTINO

8. 8
 identify and illustrate parts of a
circle; and
 apply theorems in solving
problems in tangent, secant
segments, arcs, chords and angles
of a circle.

9. CIRCLE
9
It is the set of all
points on a plane at a
given distance from a
fixed point called the
center.

10. 10
 A circle is named by
its center.
Example: ʘA A
center

11. 11

12. RADIUS
12
It is any segment
joining the center to a
point on
the circle.
radius
A B
Example: AB

13. 13
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

14. 14
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

15. 15
Congruent Circles
It composed of two or more
circles having congruent radii.
Example: ʘA ≅ʘZ
A
Z

16. 16
Concentric Circles
It composed of two or more
coplanar circles having the same
center.

17. CHORD
17
It is a segment
joining any two
points on
the circle. chord
E F
Example: EF

18. DIAMETER
18
It is a chord passing
through the center. It
twice the
measure of
the radius.
diameter
H
G
Example: GH

19. 19
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

20. ARC
20
It is a part of a
circle between two
points on the circle.
I
J
Example: IJ

21. 21
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

22. 22
Major Arc
It is an arc of a circle
having a measure greater
than 180°.
Example: OPQ
O
P Q

23. 23
Minor Arc
It is an arc smaller than a
semicircle. It measures less
than 180°.
Example: QO
O
P Q

24. 24
Arc Addition Postulate
The measure of an arc
formed by two adjacent non-
overlapping arcs is the sum
of the measures of those arcs.

25. 25
Example 1:
If VR = 54⁰;RS = 43⁰
What is VRS?
Solution:
mVR+mRS = mVRS
54⁰ + 43⁰ = 97⁰
R
S
U
V

26. 26
Example 2:
If RSU =176⁰;RS = 49⁰
What is mSU? R
S
U
V

27. 27
Intercepted Arc
The arc that lies in the
interior of an angle and has
endpoints on
the angle.
Y
intercepted arc

28. ANGLE
28
Central Angle
An angle whose vertex
is the center of the circle
and whose sides are radii
of the circle.

29. 29
Example:
 Central angle and its
intercepted arc are same in
measure. Thus, m WAX = mWX
central angle
intercepted arc A
W
X

30. Theorems on Arcs,
Chords, and
Central Angles
30

31. 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

32. Theorem 2: In the same circle
or congruent circles, two minor
arcs are congruent if and only if
their chords are
congruent.
32
A
X
Y
Z

33. Theorem 3: If a diameter is
perpendicular to a chord, then it
bisects the chord and
its arc.
33
A
X
Y
Z
W

34. Theorem 4: In the same circle
or congruent circles, two chords are
equidistant from the center(s) if and
only if they are
congruent.
34
A
X
Y
Z
W
U
V

35. 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.
35
A
X
Y
Z
W
U
V

36. 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.
36
A
X
Y
Z
W
U
V

37. ANGLE
37
Inscribed Angle
An angle whose vertex is
on a circle and whose
sides contain chords of
the circle.

38. 38
Inscribed Angle Theorem
The measure of an angle is half the
measure of its intercepted arc.
inscribed angle
intercepted arc
Y
W
X

39. 39
Solution:
½ WX= WXY
2 (½ WX= 45⁰ ) 2
WX = 90⁰
Y
W
X
45⁰
Example:
What is WX?

40. 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

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

42. Corollary 3:
If an inscribed angle
intercepts a
semi-circle, then
the angle is a
right angle.
42
C
D
B
A E

43. Corollary 4:
If two arcs of a
circle are included
between parallel
segments, then the
arcs are congruent.
43
E
C
D
B
A

44. SECANT
44
It is any line, ray, or
segment that intersects
circle in two
points.
J
I
Example: IJ

45. TANGENT
45
It is a line that
intersects at exactly one
point on a circle. The
point is called the point
of tangency.

46. 46
Example:
BC is a tangent
BC is a tangent ray
C is the point
of tangency
BC is a tangent segment
B
C

47. 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.
47
B
C

48. Corollary 1: Two tangent
segments from a common external
point are congruent.
48
A
C
B
D

49. 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.
49
A
C
B
D

50. Theorems on
Tangent, Secants
and Angle
50

51. 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
51
B A
C
D
E

52. 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

53. 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
53
B
A
C
D
E

54. Theorems on
Circles and
Segment Lengths
54

55. 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

56. 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

57. 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.
57

58. 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

59. 59

60. 60
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:

61. 61
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:

62. 62
Seatwork:
C
x
B
A
80⁰
D
9.
E

63. 63
Seatwork:
C
x A
65⁰
D
10. 11.
x
16⁰
78⁰

64. 64
Seatwork:
C
53⁰
D
12. 13. x
4
5
x
99⁰
62⁰

65. 65
Seatwork:
C
D
14. 15. x 6
12
x
8

66. “
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