PLease only complete numbered problems:
p463 # 1,2,5,6,10,11,13,14,16 – DUE ASAP,
p 473 # 1-15 due later
Ok…these two has the questions…please do page 463 ones first..page number on bottom of pages
Angle-Arc Summar
y
Central Angle Chord-Chord Angle
:0
~
( 15;-
___a.ogent Angle
.i-> S
Secant-Tangent.
p~d
B C
T
1 ——–…..—-.. 1 ~…..—-.. 1 …..—-..
mLP = 2(mCD – mAB) mLP = 2(mSXT – mST) mLP = 2(mRT –
Vertex outside circle ~ half the difference
•
Find y.
Find mLBEC first.
mLBEC = ~(29 + 47) = 38
Thus, y = 180 – mLBEC = 142.
Part Two: Sample Problems
Problem 1
104′
Given: AB is a diameter of OP. ~E
ED = 20°, DE = 104° C 20°
A
Find:mLC ~
First find mEA.
~ …..—-..
mAEB = 180, so mEA = 180 – (104 + 20) = 56.
1″”‘—-” …..—-.. 1
Thus, mLC = 2(mEA – mDB) = 2(56 – 20) = 18.
Solution
Problem 2
Solution
472 Chapter 10 Circles
Problem 3
Solution
Problem 5
Solution
a Find x. b Find y.
Solution a x = ~(88 + 27) b y = ~(57 – 31) c z = ~(233 – 127)= 57.! = 13 = 532
Problem 4 a Find y. b Find z. c Find a.
Part Three: Problem Sets
Problem Set A
1 Vertex at center:
Given: AB = 62°
Find: mLO
a ~(21 + y) = 72
21 + Y = 144
Y = 123
b ~(125 – z) = 32
125 – z = 64
z = 61
Find mAB and mED. ../
Let mAB = x arid mED = y.
Then ~(x + y) = 65 and ~(x – y) = 24.
So x + Y = 130 and x – y = 48.
x + Y = 130
x – Y = 48
2x = 178 Add the equations.
X = 89
89 + Y = 130
Y = 41
Thus, mAB = 89 and mED = 41.
c Find z.
233
c ~a = 65
a = 130
F
Section 10.5 Angles Related to a Circle 473
Problem Set A, continued c
2 Vertex inside:
Given: CD = 100°, Fe = 30°
Find: mLCED
3 Vertex on:
a Given: AC = 70°
Find: mL
B
F
B
Db Given: DE is tangent at E.EF = 150°
Find: mLDEF
E
4 Vertex outside:
ba c
w
R
K R
Given: Wand R are points of
contact.
WR = 140°
Find: mLX
T
Given: HP = 120°,
AM = 36°
Find: mLK
Given: TU is tangent at U.
RD = 160°,
§D = 60°
Find: mLT
5 Find the measure of each angle or arc that is labeled with a letter.
160’c ea
x
11
10
..—….
b d
12 1-,
120c 810
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
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
= 23
See problems 16, 21, 22, and 29 for other types of
walk-around problems.
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.
A
B x 15 – x ,c
15 – x
(10 – x)
10 x
A 10 – x
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.)
3 Given: PR and PQ are tangents to 00 at
Rand Q.
—–7 _
Prove: PO bisects LRPQ. (Hint: Draw RO
and OQ.)
4 Given: AC is a diameter of OB.
Lines sand m are tangents to the
o at A and C.
Conclusion: s II m
x
y
p~ –=-…’.R-,–
Q
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
6 AB and AC are tangents to 00,
and OC = 5x. Find OC.
B
A~
19 – 6x C
7 Given: CE is a common internal tangent
to circles A and B at C and E.
Prove: a LA == LB
b AD = CD
BD 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.)
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.)
Note This is part of the proof of a useful
property: The common external tangent
segments of two circles are congruent.
Problem Set B
10 OP is tangent to each side of ABCD.
AB = 20, BC = 11, and DC = 14. Let
AQ = x and find AD.
464 Chapter 10 Circles
w
p
z
A
x-axis
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
points Sand T.
. PQ SQ
Conclusion: PR = 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 se-
cant segment.
b Find OP.
T
P
Section 10.4 Secants and Tangents 465
