CIRCLES IN MATHEMATICS 10 EMERALD CANAN NHS

1. Circles

2. Questions
 How do I identify segments and lines
related to circles?
 How do I use properties of a tangent
to a circle?

3. Definitions
 A circle is the set of all points in a plane that are
equidistant from a given point called the center of the
circle.
 Radius – the distance from the center to a point on the
circle
 Congruent circles – circles that have the same radius.
 Diameter – the distance across the circle through its
center

4. Diagram of Important Terms
diameter
radius
P
center
name of circle: P

5. Definition
 Chord – a segment whose endpoints are
points on the circle.
AB is a chord
B
A

6. Definition
 Secant – a line that intersects a circle in
two points.
MN is a secant
N
M

7. Definition
 Tangent – a line in the plane of a circle
that intersects the circle in exactly one
point.
ST is a tangent
S
T

8. Example 1
 Tell whether the line or segment is best described
as a chord, a secant, a tangent, a diameter, or a
radius.
F
C
B
G
A
H
D
E
I
d. CE
c. DF
b. EI
a. AH tangent
diameter
chord
radius

9. Definition
 Tangent circles – coplanar circles
that intersect in one point

10. Definition
 Concentric circles – coplanar circles
that have the same center.

11. Definitions
 Common tangent – a line or segment that is
tangent to two coplanar circles
 Common internal tangent – intersects the segment
that joins the centers of the two circles
 Common external tangent – does not intersect the
segment that joins the centers of the two circles
common external tangent
common internal tangent

12. Example 2
 Tell whether the common tangents are internal or
external.
a. b.
common internal tangents common external tangents

13. More definitions
 Interior of a circle – consists of
the points that are inside the
circle
 Exterior of a circle – consists of
the points that are outside the
circle

14. Definition
 Point of tangency – the point at which a
tangent line intersects the circle to which
it is tangent
point of tangency

15. Perpendicular Tangent
Theorem
 If a line is tangent to a circle, then it is
perpendicular to the radius drawn to the
point of tangency. l
Q
P
If l is tangent to Q at P, then l  QP.

16. Perpendicular Tangent
Converse
 In a plane, if a line is perpendicular to a
radius of a circle at its endpoint on the
circle, then the line is tangent to the
circle. l
Q
P
If l  QP at P, then l is tangent to Q.

17. Definition
 Central angle – an angle whose vertex is the
center of a circle.
central angle

18. Definitions
 Minor arc – Part of a circle that
measures less than 180°
 Major arc – Part of a circle that
measures between 180° and 360°.
 Semicircle – An arc whose endpoints
are the endpoints of a diameter of the
circle.
Note : major arcs and semicircles are
named with three points and minor
arcs are named with two points

19. Diagram of Arcs
C
D B
A
minor arc: AB
major arc: ABD
semicircle: BAD

20. Definitions
 Measure of a minor arc – the
measure of its central angle
 Measure of a major arc – the
difference between 360° and the
measure of its associated minor arc.

21. Arc Addition Postulate
 The measure of an arc formed by two adjacent
arcs is the sum of the measures of the two arcs.
A
C
B
mABC = mAB + mBC

22. Definition
 Congruent arcs – two arcs of the same
circle or of congruent circles that have
the same measure

23. Arcs and Chords Theorem
 In the same circle, or in congruent circles, two
minor arcs are congruent if and only if their
corresponding chords are congruent.
A
B
C
AB BC if and only if AB BC

24. Perpendicular Diameter
Theorem
 If a diameter of a circle is perpendicular to a
chord, then the diameter bisects the chord and
its arc.
D
F
G
E
DE EF, DG FG

25. Perpendicular Diameter
Converse
 If one chord is a perpendicular bisector of
another chord, then the first chord is a
diameter.
L
M
J
K
JK is a diameter of the circle.

26. Congruent Chords Theorem
 In the same circle, or in congruent circles, two
chords are congruent if and only if they are
equidistant from the center.
G
F
E
C
D
B
A
AB CD if and only if EFEG.

27. Right Triangles
Pythagorean Theorem
45
43
11
D
E
C
Radius is perpendicular to the
tangent.  < E is a right angle

28. Example 3
Tell whether CE is tangent to D.
45
43
11
D
E
C
Use the converse of the Pythagorean
Theorem to see if the triangle is right.
112
+ 432
? 452
121 + 1849 ? 2025
1970  2025
CED is not right, so CE is not tangent to D.

29. Congruent Tangent Segments
Theorem
 If two segments from the same exterior
point are tangent to a circle, then they
are congruent.
S
P
R
T
If SR and ST are tangent to P, then SR ST.

30. Example 4
AB is tangent to C at B.
AD is tangent to C at D.
Find the value of x.
11
x2 + 2
A
C
D
B
AD = AB
x2 + 2 = 11
x2 = 9
x = 
3

31. Example 1
 Find the measure of each arc.
70
P
N L
M
a. LM
c. LMN
b. MNL
70°
360° - 70° = 290°
180°

32. Example 2
 Find the measures of the red arcs. Are the arcs congruent?
41
41
A
C
D
E
mAC = mDE = 41
Since the arcs are in the same circle, they are congruent!

33. Example 3
 Find the measures of the red arcs. Are the arcs congruent?
81
C
A
D
E
mDE = mAC = 81
However, since the arcs are not of the same circle or
congruent circles, they are NOT congruent!

34. Example 4
A
(2x + 48)
(3x + 11)
D
B
C
3x + 11 = 2x + 48
Find mBC.
x = 37
mBC = 2(37) + 48
mBC = 122

35. Definitions
 Inscribed angle – an angle whose vertex is
on a circle and whose sides contain chords
of the circle
 Intercepted arc – the arc that lies in the
interior of an inscribed angle and has
endpoints on the angle
inscribed angle
intercepted arc

36. Measure of an Inscribed Angle Theorem
 If an angle is inscribed in a circle, then its
measure is half the measure of its
intercepted arc.
C
A
D B
mADB =
1
2
mAB

37. Example 1
 Find the measure of the blue arc or angle.
R
S
Q
T
a.
mQTS = 2(90) = 180
b.
80
E
F
G
mEFG =
1
2
(80) = 40

38. Congruent Inscribed Angles Theorem
 If two inscribed angles of a circle intercept the same
arc, then the angles are congruent.
A
C
B
D
C D

39. Example 2
It is given that mE = 75. What is mF?
D
E
H
F
Since E and F both intercept
the same arc, we know that the
angles must be congruent.
mF = 75

40. Definitions
 Inscribed polygon – a polygon whose
vertices all lie on a circle.
 Circumscribed circle – A circle with an
inscribed polygon.
The polygon is an inscribed polygon and
the circle is a circumscribed circle.

41. Inscribed Right Triangle
Theorem
 If a right triangle is inscribed in a circle, then
the hypotenuse is a diameter of the circle.
Conversely, if one side of an inscribed
triangle is a diameter of the circle, then the
triangle is a right triangle and the angle
opposite the diameter is the right angle.
A
C
B
B is a right angle if and only if AC
is a diameter of the circle.

42. Inscribed Quadrilateral
Theorem
 A quadrilateral can be inscribed in a circle
if and only if its opposite angles are
supplementary.
C
E
F
D
G
D, E, F, and G lie on some circle, C if and only if
mD + mF = 180 and mE + mG = 180.

43. Example 3
 Find the value of each variable.
2x
Q
A
B
C
a.
2x = 90
x = 45
b. z
y
80
120
D
E
F
G
mD + mF = 180
z + 80 = 180
z = 100
mG + mE = 180
y + 120 = 180
y = 60

44. Tangent-Chord Theorem
 If a tangent and a chord intersect at a
point on a circle, then the measure of
each angle formed is one half the
measure of its intercepted arc.
2
1
B
A
C
m1 =
1
2
mAB
m2 =
1
2
mBCA

45. Example 1
m
102
T
R
S
Line m is tangent to the circle. Find mRST
mRST = 2(102)
mRST = 204

46. Try This!
Line m is tangent to the circle. Find m1
m
150
1
T
R
m1 =
1
2
(150)
m1 = 75

47. Example 2
(9x+20)
5x
D
B
C
A
BC is tangent to the circle. Find mCBD.
2(5x) = 9x + 20
10x = 9x + 20
x = 20
mCBD = 5(20)
mCBD = 100

48. Interior Intersection Theorem
 If two chords intersect in the interior of a
circle, then the measure of each angle is
one half the sum of the measures of the
arcs intercepted by the angle and its
vertical angle.
m1 =
1
2
(mCD + mAB)
m2 =
1
2
(mAD + mBC)
2
1
A
C
D
B

49. Exterior Intersection
Theorem
 If a tangent and a secant, two tangents, or two secants
intersect in the exterior of a circle, then the measure of
the angle formed is one half the difference of the
measures of the intercepted arcs.

50. Diagrams for Exterior
Intersection Theorem
1
B
A
C
m1 =
1
2
(mBC - mAC)
2
P
R
Q
m2 =
1
2
(mPQR - mPR)
3
X
W
Y
Z
m3 =
1
2
(mXY - mWZ)

51. Example 3
 Find the value of x.
174
106
x
P
R
Q
S
x =
1
2
(mPS + mRQ)
x =
1
2
(106+174)
x =
1
2
(280)
x = 140

52. Try This!
 Find the value of x.
120
40
x
T
R
S
U
x =
1
2
(mST + mRU)
x =
1
2
(40+120)
x =
1
2
(160)
x = 80

53. Example 4
 Find the value of x.
200
x
72
72 =
1
2
(200 - x)
144 = 200 - x
x = 56

54. Example 5
 Find the value of x.
mABC = 360 - 92
mABC = 268 x
92
C
A
B
x =
1
2
(268 - 92)
x =
1
2
(176)
x = 88

55. Chord Product Theorem
 If two chords intersect in the interior of a
circle, then the product of the lengths of
the segments of one chord is equal to the
product of the lengths of the segments of
the other chord.
E
C
D
A
B
EA EB = EC ED

56. Example 1
 Find the value of x.
x
9
6
3
E
B
D
A
C
3(6) = 9x
18 = 9x
x = 2

57. Try This!
 Find the value of x.
x 9
18
12
E
B
D
A
C
9(12) = 18x
108 = 18x
x = 6

58. Secant-Secant Theorem
 If two secant segments share the same
endpoint outside a circle, then the product of
the length of one secant segment and the
length of its external segment equals the
product of the length of the other secant
segment and the length of its external
segment.
C
A
B
E
D
EA EB = EC ED

59. Secant-Tangent Theorem
 If a secant segment and a tangent segment
share an endpoint outside a circle, then the
product of the length of the secant segment
and the length of its external segment equals
the square of the length of the tangent
segment.
C
A
E
D
(EA)2 = EC ED

60. Example 2
 Find the value of x.
LM LN = LO LP
9(20) = 10(10+x)
180 = 100 + 10x
80 = 10x
x = 8 x
10
11
9
O
M
N
L
P

61. Try This!
 Find the value of x.
x
10
12
11
H
G
F
E
D
DE DF = DG DH
11(21) = 12(12 + x)
231 = 144 + 12x
87 = 12x
x = 7.25

62. Example 3
 Find the value of x.
x
12
24
D
B
C
A
CB2 = CD(CA)
242 = 12(12 + x)
576 = 144 + 12x
432 = 12x
x = 36

63. Try This!
 Find the value of x.
3x
5
10
Y
W
X Z
WX2 = XY(XZ)
102 = 5(5 + 3x)
100 = 25 + 15x
75 = 15x
x = 5

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