Geometry Semester 2
Homework on Circles and Segment Lengths
1.
Name: ___________________________
Give the missing reasons for the following
proof that a line drawn through the center
of a circle perpendicular to a chord meets it
at its midpoint.
O
(That is, if OP  AB then N is the
midpoint of AB .)
A
B
N
P
Proof:
Statement:
2.
Reason:
1.
OP  AB
Given
2.
ANO and BNO are right s
Definition of perpendicular
3.
ANO and BNO are right s
Definition of right triangle
4.
ON  ON
Reflexive
5.
OA  OB
6.
AON  BON
7.
AN  BN
8.
N is the midpoint of AB
Definition of midpoint
In circle Q, QC  AB and AC = 5.
What is the length of AB ?
B
Q
C
A
Page 1 of 8
3.
P
The radius of circle K is 5 cm, PR  KM
at Q, and PR  6 cm . What is the length of KQ ?
Q
M
K
R
4.
Fill in the missing reasons for the following
proof that if two circles intersect, the line joining
their centers is the perpendicular bisector of the
common chord.
P
A
N
(That is, prove that PN  AB and PN = QN.)
B
Q
Hint: We first show triangles APB and AQB are congruent. Then we show triangles APN and
AQN are congruent.
Proof:
Statement:
Reason:
1.
AP  AQ
These are both radii of circle A
2.
BP  BQ
3.
AB  AB
4.
APB  AQB
5.
AP  AQ
Listed in step 1 (or CPCTC)
6.
AN  AN
Reflexive
7.
PAN  QAN
8.
APN  AQN
9.
PNA  QNA
10.
PN  QN
CPCTC
11.
PNA and QNA form a line
Given in diagram
12.
PNA  QNA  180o
A line has 180o
13.
PNA  90o and QNA  90o
Using algebra on steps 9 and 12
14.
PN  AB
15.
AB is the  bisector of PQ
Definition of perpendicular bisector, referring to
steps 10 and 14.
Page 2 of 8
5.
A
The radius of circle P is 10 inches, the radius
of circle Q is 8 inches, and PC = 8 inches.
What is the length of AB ?
P
Q
B
6.
Fill in the missing reasons for the
Q
following proof that two tangents drawn to
a circle from an external point are equal,
and make equal angles with the line joining
P
O
the point to the center of the circle.
(We prove that if PQ and PR are tangent
R
to circle O, then PQ = PR and angles QPO
and RPO are equal.)
Hint: Since a line tangent to a circle is perpendicular to the radius at the point of tangency, triangles
PQO and PRO are right triangles, and they are congruent.
Proof:
Statement:
1.
PQ and PR are tangent to circle
O at points Q and R
2.
PQ  OQ and PR  OR
3.
PQO and PRO are right s
4.
PO  PO
5.
OQ  OR
6.
PQO  PRO
7.
PQ = PR
8.
QPO  RPO
Reason:
Given
A line tangent to a circle is perpendicular to the
radius at the point of tangency.
Definition of right triangle
Page 3 of 8
7.
A circle of radius 20 in. is externally
tangent to a circle of radius 5 in. Find the
length of a common external tangent.
(That is, if OX = 20 in. and PY = 5 in., find
XY.)
Hints: OX  XY and PY  XY . Draw
NP  OX . Then NP = XY and NP can be
found by the Pythagorean Theorem.
8.
In circle O, OP  AP , AB = 12 and
OP = 8. What is the radius, OB?
O
A
9.
B
P
AC is a common external tangent of circles O and P, AC = 24, and OC = 26.
A
B
C
N
P
O
a.
Use the Pythagorean Theorem to find the radius of circle O (that is, find OA).
b.
How long is CN ?
c.
Let PB = x. Then PN = x since it is also a radius of circle P. How long is PO in terms of x ?
d.
How long is CP in terms of x?
e.
Use the fact that CBP ~ CAO to find x.
Page 4 of 8
10. In the following diagram two chords intersect to form segments of lengths a, b, x, and y:
a
y
b
x
We draw two more chords and label endpoints in order to use similar triangles:
Q
y
a
P
x
T
b
S
R
(a) RPS and RQS both intercept the same arc, so they are equal. Likewise, PRQ and
PSQ both intercept arc PQ so they too are equal. Therefore, by the AA theorem,
 PRT ~  _______.
(b) So these triangles are similar, with side x of  PRT corresponding to side b of  QST and
side a of  PRT corresponding to side y of  QST . Therefore we can equate ratios:
a
x

b
(c) Now cross-multiply: ______ = _______
Problems 11-12: Find the value of x.
11.
x
2
4
12.
x+5
6
x
x+2
x–2
Page 5 of 8
13. Consider a circle with two secants from a common external point:
a
b
y
x
We draw two other segments and label points so we can again consider similar triangles:
Q
a
P
b
T
y
S
x
R
(a) TPS  TRQ . Why? _________________________________________________
(b) By the reflexive property, T  T , so  TPS ~  _____________
(c) Using corresponding sides of  TPS and  TRQ ,
ab

x y
y
(d) Cross multiply to get: b(a  b)  ____________
Problems 14-16: Find the value of x.
12
14.
9
15.
x
9
6
x
12
Page 6 of 8
16.
x
O
8
9
15
17.
Given:
P with radius 5 cm,
RS = 8 cm,
QS is tangent to
P
P at Q
R
Find QS
18.
If AB is tangent to circle O at point T,
250
then what is the measure of angle CTB?
S
tangent line
Q
o
C
O
T
A
19.
RS is tangent to circles P and Q. The
radius of
of
B
R
P is 15 inches, the radius
S
Q is 8 inches, and these circles are
2 inches apart. How long is RS?
P
Page 7 of 8
15
2
8
Q
Problems 20-22:
Given: AB  CD
A
mAP = 60o,
radius of
O = 5,
P
RB = 4
20.
What is mABP ?
O
21.
Find the exact value of RS.
22.
Find the exact value of PB.
C
R
S
B
Page 8 of 8
D