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
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
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
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
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
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
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:
