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Angle-Arc Summary
Central Angle
Chord-Chord Angle
~
:0
(
___
a.ogent Angle
.i->
p~d
B
15;-
Secant-Tangent.
S
C
T
1
--------.....----..
1
mLP = 2(mCD - mAB)
~.....----..
Vertex outside circle ~
•
472
104'
Given: AB is a diameter of OP.
ED = 20°, DE = 104°
Find:mLC
~E
C
20°
~
Solution
First find mEA.
~
.....----..
mAEB = 180, so mEA = 180 - (104 + 20) = 56.
1""'----"
.....----..
1
Thus, mLC = 2(mEA - mDB) = 2(56 - 20) = 18.
Problem 2
Find y.
Solution
Find mLBEC first.
+
mLBEC
=
~(29
Thus, y
=
180 - mLBEC
Chapter 10 Circles
47)
.....----..
mLP = 2(mRT -
half the difference
Part Two: Sample Problems
Problem 1
1
mLP = 2(mSXT - mST)
=
38
=
142.
A
Problem
3
a Find x.
b Find y.
c Find z.
233
Solution
a x
= ~(88 +
= 57.!
27)
b y
2
Problem 4
a Find y.
Solution
a ~(21
21
Problem 5
Solution
+ y) =
+Y =
Y =
72
144
123
X
89
+Y
130
=
48
= 178
= 89
=
=
=
~(233 - 127)
53
c Find a.
=
=
z=
32
64
61
c ~a
a
= 65
=
130
../
=
y.
Then ~(x + y) = 65 and ~(x - y)
So x + Y = 130 and x - y = 48.
=
c z
13
b ~(125 - z)
125 - z
mAB and mED.
Let mAB = x arid mED
2x
~(57 - 31)
b Find z.
Find
x +Y
x - Y
=
=
=
24.
F
Add the equations.
130
Y = 41
Thus,
mAB
= 89 and
mED
=
41.
Part Three: Problem Sets
Problem Set A
1 Vertex at center:
Given:
AB =
62°
Find: mLO
Section 10.5
Angles Related to a Circle
473
c
Problem Set A, continued
2 Vertex inside:
Given: CD = 100°,
Fe =
30°
Find: mLCED
3 Vertex on:
a Given: AC
Find: mLB
=
70°
F
B
b Given: DE is tangent at E.
EF =
D
150°
Find: mLDEF
E
4 Vertex outside:
a
w
c
b
x
R
K
Given:
HP =
AM
=
120°,
36°
R
T
Given: TU is tangent at U.
RD
=
§D =
Find: mLK
160°,
60°
Given: Wand R are points of
contact.
WR =
Find: mLT
10
140°
Find: mLX
5 Find the measure of each angle or arc that is labeled with a letter.
c
a
e
160'
11
..---....
d
b
12 1
-,
120c
0
81
82c
474
Chapter 10
Circles
Problem 4
A walk-around problem:
Given: Each side of quadrilateral
ABCD is tangent to the circle.
AB = 10, BC = 15, AD = 18
Find: CD
A
Solution
Let BE = x and "walk around" the
figure, using the given information
and the Two-Tangent Theorem.
CD = 15 - x + 18 - (10 - x)
= 15 - x + 18 - 10 + x
=
B
x
15 - x
,c
15 - x
(10 - x)
23
10
See problems 16, 21, 22, and 29 for other types of
walk-around problems.
x
A
10 - x
Part Three: Problem Sets
Problem Set A
1 The radius of OA is 8 cm.
Tangent segment BC is 15 cm long.
Find the length of AC.
2 Concentric circles with radii 8 and 10
have center P.
XY is a tangent to the inner circle and is
a chord of the outer circle.
Find XY. (Hint: Draw PX and PY.)
x
y
3 Given: PR and PQ are tangents to 00 at
Rand Q.
-----7
_
Prove: PO bisects LRPQ. (Hint: Draw RO
and OQ.)
--=-...'.R-,--
p~
Q
4 Given: AC is a diameter of OB.
Lines sand m are tangents to the
o at A and C.
Conclusion: s II m
Section 10.4 Secants and Tangents
463
Problem Set A, continued
5 OP and OR are internally tangent at O.
P is at (8, 0) and R is at (19, 0).
a Find the coordinates of Q and S.
b Find the length of QR.
o
x-axis
B
6 AB and AC are tangents to 00,
and OC = 5x. Find OC.
A~
19 - 6x
C
7 Given: CE is a common internal tangent
to circles A and B at C and E.
Prove: a LA
b AD
== LB
=
BD
CD
DE
8 Given: QR and QS are tangent to OP at
points Rand S.
Prove: PQ 1. RS (Hint: This can be
proved in just a few steps.)
w
9 Given: PW and PZ are common tangents
to
@
A and B at W, X, Y, and Z.
Prove: WX == YZ (Hint: No auxiliary
lines are needed.)
p
Note This is part of the proof of a useful
property: The common external tangent
segments of two circles are congruent.
z
Problem Set B
10 OP is tangent to each side of ABCD.
AB = 20, BC = 11, and DC = 14. Let
AQ
464
= x and find AD.
Chapter 10
Circles
A
11 a Find the radius of OP.
b Find the slope of the tangent to OP at
point Q.
x-axis
12 Two concentric circles have radii 3 and 7. Find, to the nearest
hundredth, the length of a chord of the larger circle that is
tangent to the smaller circle. (See problem 2 for a diagram.)
13 The centers of two circles of radii 10 em and 5 cm are 13 em
apart.
a Find the length of a common external tangent. (Hint: Use the
common -tangent procedure.)
b Do the circles intersect?
14 The centers of two circles with radii 3 and 5 are 10 units apart.
Find the length of a common internal tangent. (Hint: Use the
common-tangent procedure.)
15 Given: PT is tangent to ®
Q and R at
T
points Sand T.
. PQ
Conclusion: PR
SQ
=
TR
P -====-------i-*--l---+--------!
16 Given: Tangent ® A, B, and C,
AB = 8, BC = 13, AC = 11
Find: The radii of the three ® (Hint:
This is a walk-around problem.)
17 The radius of 00 is 10.
The secant segment PX measures 21 and
is 8 units from the center of the O.
a Find the external part (PY) of the secant segment.
P
b Find OP.
Section 10.4
Secants and Tangents
465