Download Chapter 13 - Haiku Learning

Chapter 13
At carnivals around the world people riding on ferris wheels get a bird's eye view of
the fairgrounds. The invention of the wheel, essential to transportation and industry,
was critical to the development of modem civilization.
379
Chapter
13
Focus on· Skills
Multiply.
1.
1
2. 182
1
2 . 47
2.
3.
21 (58+ 112)
4.
~ (220
- 140)
GEOMETRY
Draw each figure described. Construct the indicated line or ray.
5. AB; the perpendicular bisector of AB
6. line m and Q on m; the perpendicular to m at Q
7. any acute angle DEF; the bisector of LDEF
In Exercises 8 and 12 use 3.14 for 1t.
8. The diameter of a circle is 20 em. Find (a) the radius, (b) the
circumference, and (c) the area of the circle.
9. Complete: 1 turn = •
o
10. What fractional part of
the circle is each section
shown?
11. Find the measure of LRPT
in amount of tum and in
degrees.
0
0
12. Find the area of the
shaded region.
T
13. Find the area of a triangle with sides of length 15m, 12m, and 9 m.
c:9
Complete and give a reason for each statement.
14.
15.
16.
X
p
w
D
a. LAED
:= .
b. h.AED- .
, c. AE = DE
• •
380
c
s
y
a. LW
a.LP= •
b. h.PRT-
c.
Chapter 13 Focus on Skills
PR = PT
•
:= .
b. h.WXY- .
c. WY = WX
•
13.1
Arcs and Central Angles
Objective: To determine the measures of arcs of circles.
A diameter of a circle divides the circle into two semicircles. In the
figure shown, AcE (read as "arc ACE") and Ai5ii are semicircles ofOO.
r
Minor arcs are shorter than semicircles.
Major arcs are longer than semicircles.
~inorarcs:AC, CB,EUJ,~,1JC
~ajor arcs:
Ciili, BOC, J5CB, ACi5, CBA
Notice that we name a minor arc by its endpoints. We use
three points to name semicircles and major arcs.
D
The measure of a semicircle is 180°. The measure of a minor arc is defined
as the measure of its central angle. The measure of a major arc is 360°
minus the measure of the associated minor arc.
Example: ForOO, mL.XOY = 110°. Find each of the following.
a.mXY
b. mXYZ
y
c. mfZX
d. mYZ
Solution: a. mXY = 110°
c. mfZX = 250°
b. mXYZ = 180°
d. mYZ = 70°
Y:Jj)x
EXPLORING
00 has radius 1 em. The circles with center P have radii 1 em and 1.5 em.
1. N arne the arcs shown that have measure 60°.
2. The length of an arc is a fractional part of the
circumference of a circle. a. What fractional
part of the circle is GH? EF? b. Find the
lengths of Gil and EF. c. ~ust arcs that have
the same measure also have the same length?
D
3. How would you define congruent arcs?
13.1 Arcs and Central Angles
381
Circles that have the same radii are called congruent circles. The
EXPLORING activity suggests the following definition and theorem.
Congruent arcs are arcs of the same circle (or congruent circles) that have
the same measure.
THEOREM 13.1: In the same circle (or congruent circles), two
minor arcs are congruent when their central angles are congruent.
. ,Thin I< I119 · e:rl'iJcal' l'v~>g%·'@fi·E~
1. Suggest a formula for the length l of an arc in terms of the measure n°
of the arc and the radius r of the circle.
2. The shaded region is called a sector of the circle.
a. How could you find the area of the shaded region?
b. If the radius of the circle is 3 em and the measure of
the arc is 80°, find the area of the shaded region.
G n"
Class Exercises
1. Name each of the following for Oo.
a. two minor arcs
b. two major arcs
c. a semicircle
2. Find each measure.
a. mLYOZ
b.
d.mXYZ
e.
mXY
ml'ZX
c.
f.
miZ
mYXZ
G8y
z
Exercises
1. Name each of the following for OQ.
a. two minor arcs
b. two major arcs
2. Find each measure.
a. mLLQC
d.mCIJ5
b. mLLQP
e. mLPC
c. a semicircle
c. mLP
f. mLcP
3. What fractional part of a circle is an arc with the given measure?
a. 30°
382
Chapter 13
b. 45°
Circles
c. 120°
(D
L
soo c
Find each indicated measure for 00.
=.
4. mLAOB = 45°
mAB
6.
mXY
b.
mLCOD
=.
= 110°
a. mLXOY
mXAY
7.
A
= •
8. a. mL3
mXB = 55°
a. mAX= •
b. mARX= •
B
A
9. a. mLROS
b. mBC
b. mST
c. mL1
c. mPT
d. mAD
d. mSP
e. mL2
e. mRsT
f.
=.
5. mCD = 120°
mA-cE
I
1\
I
-
~R
I 6QC
f. mPRT
10. a. mLAOB
11. a. mLXOY
b. mBC
b. mLYOZ
c. mLAOC
c. mLXOZ
d. mAC
d. mXZ
e. mADC
e. mXWZ
f. mBcA
f. mYWZ
. .---!---.. '
~
12. The three circles have center 0. Each circle is divided into
sixteen congruent arcs.
a. mLAOB
=•
=•
=•
b. mLCOD
c. mLEOF
f. mEF = •
d. mAB = [I
e. mCi5
g. Are the arcs on the largest circle the same length as the
arcs on the smaller circles?
=.
13. Draw two congruent circles.
14. Draw two circles that have center X and radii 2 em and 5 em.
13.1 Arcs and Central Angles
383
For each exercise draw a circle having an arc or arcs with the indicated
measure(s).
16. mCDE = 210°
15. mAE= 75°
17. mFG- = 80°, mJK = soo
18. mXY = 110°, mY2 = 70°
19. mRS = 100°, mTW = 35°
20.
mAE =
125°,
mARC = 300°
21. The radius of a circle is 14 in. Find the length of an arc whose measure
is 45°. (Use 2 for 7t.)
22. The radius of the circle shown at the right is 6 em. Find the area of the
shaded region. (Use 3.14 for 7t.)
i
23. The circumference of a circle is 727t em. If the length of one arc
of the circle is 497t em, what is the measure of the arc?
APPLICATIONS
Engineering The positions of the five lugs on an automobile wheel
could be located by drawing five congruent central angles. For each
exercise draw a circle with a radius of at least 5 em. Locate the given
number of equally-spaced points around the circle.
24. 5 points
25. 3 points
26. 6 points
27. 8 points
Historical Note
About 200 B.C. Eratosthenes, a Greek geographer, estimated the
circumference of Earth. His calculations were based on assumptions
that Earth is round and that since the sun is so far from Earth, the
sun's rays are virtually parallel when they reach Earth.
Eratosthenes noticed that in Syene (S}, a town
south of Alexandria, Egypt, the sun was directly
overhead at noon on a certain day. When the sun
was directly overhead in Syene, he measured
LOAB, the angle formed by a vertical post in
Alexandria (A) and the tip of its shadow. The angle
was 5 ~ of a circle. Thus Eratosthenes also knew the
measure of LAOS.
1. Why were LOAB and LAOS congruent?
2. Eratosthenes knew the distance between Alexandria
and Syene. How did he estimate the circumference of Earth?
384
Chapter 13
Circles
ce
13.2
Inscribed Angles
Objective: To use the relationship between the measures of an inscribed
angle and its intercepted arc.
An inscribed angle of a circle is an angle with vertex on the circle and
sides containing chords of the circle. In the figure at the right, LRST
is an inscribed angle. 7ft is its intercepted arc.
kJs
EXPLORING
1. Use the Geometric Supposer: Circles disk to draw a circle
with radius 5. Label moveable points B, C, D, and Eon the
circle. Position point E so that liCE is a major arc.
B
c
2. Draw CB, CE, DB, DE, AB, andAE.
3. Measure the ratio of mLBCE to mLBAE.
Measure the ratio of mLBDE to mLBAE.
4. What is the ratio of the measure of each inscribed angle to
mBE?Why?
I
The EXPLORING activity demonstrates the following theorem and its
corollary for the special case of an inscribed angle that intercepts a semicircle.
I
THEOREM 13.2: The measure of an inscribed angle is equal to
half the measure of its intercepted arc.
~
COROLLARY:
right angle.
An inscribed angle that intercepts a semicircle is a
Thinking Critically
1. If Ci5 is a minor arc of OP, how
many central angles intercept Ci5?
How many inscribed angles
intercept CD?
2. If two inscribed angles of a circle
intercept the same arc or congruent
arcs, what must be true of the
inscribed angles?
13.2 Inscribed Angles
385
Class Exercises
For 0 C find each measure.
1. mXY
2. mLWCX
3. mWX
4. mLZWY
5. mlVZY
6. mWZ
7. mXZ
8. mXZ¥
®X
0
z
50° y
Exercises
For each exercise draw a figure like OP. Use the points shown to draw
each of the following.
1. a central angle that intercepts AB
2. an inscribed right angle
0A
3. two inscribed angles that intercept AB
4. an obtuse central angle
5. an acute inscribed angle
6. an obtuse inscribed angle
7. a.
=.
mCFE
b. mCE
8. a.
= •
mRwT
=.
= •
b. mLRST
s
0
~a:,
-~
10. mAB = 60°
11. AX
a. mLl = •
b. mL2= •
=EX; mLl
=.
a. mL2 = •
b. mAB
9. a. mLXBY = •
b.
mXBY = •
B©!OO'
= 25°
12. a. mffij = •
b. mAD=
c. mfiQi5 = •
N
vQQ
386
Chapter 13
Circles
13. AZ is a diameter of 00.
14.
mYU = 260°, mXY = 40°
a. mAX=
b.
mABZ
c. mYZ =
d. mLXAY =
e. mLYAZ =
f.
mXAZ
= 50°,
mDC
a. mLl =
:([Y
Ill
is a diameter of 00.
mAE
A
=
Ac
b. mL2 =
=.
c. mL3 =
d. mL4
e. mL5 =
f. mAD=
z
=
15. DF is a diameter ofOA, mBF = 70°,
mCD = 50°, and mDE = goo. Find
the measure of each numbered angle.
= 80°
~
~
0
2
A
B
B
16. If all four vertices of a quadrilateral are on the same circle, what must
be true of opposite angles of the quadrilateral? Draw a diagram and use
it to explain.
17. Given a circle but not its center, describe how you could use the corner
of a sheet of paper and a straightedge to locate each of the following.
b. the center of the circle
a. a diameter
APPLICATIONS
Algebra
18.
Find the value of x.
~~
20.
19.
80 °
(6x+ 10) 0
Computer
Write LOGO procedures that draw a circle and an inscribed angle with each
of the following measures. You can use the CIRCLE procedure on page 284.
1. 45°
2. goo
3. 120°
13.2 Inscribed Angles
387
13.3
Tangents
Objective: To construct tangents and apply theorems about tangents.
A line is tangent to a circle if it is in the same plane as the circle and
intersects it in exactly one point. The point of intersection is called the
point of tangency. In the figure at the right, line l is tangent to OR at
point S. S is the point of tangency.
A line tangent to two or more circles is a common tangent to the circles.
Example:
Line tis a common internal
tangent of OP and OQ.
Line m is a common external
tangent of OR and OS.
•
[QJ
EXPLORING
1. a. Draw three or four circles and one tangent for each circle. For each
circle draw the radius to the point of tangency.
b. Describe the relationship of each radius to the tangent.
2. a. Draw three or four circles and one radius for each circle. For each
circle draw the line that is perpendicular to the radius and passes
through its endpoint on the circle.
b. Describe the relationship of each line to the circle.
The EXPLORING activity demonstrates the following theorems.
THEOREM 13.3 If a line is tangent to a circle, then the line is
perpendicular to the radius drawn to the point of tangency.
THEOREM 1 3.4: If a line is in the same plane as a circle and is
perpendicular to a radius at its endpoint on the circle, then the line is
tangent to the circle.
Construction 13 is based on Theorem 13.4.
388
Chapter 13
Circles
•m
- - - - - - - - · CONSTRUCTION 13 - - - - - - - - ·
Construct a tangent to Oo at point A on the circle.
1. Draw
OA.
2. Construct the line through A that is perpendicular to
OA.
GA
- - - - - - - - · CONSTRUCTION 14 - - - - - - - - ·
Construct the tangents to 00 from point A outside the circle.
1. Draw OA. Locate
the midpoint M of OA
by constructing the
perpendicular bisector of
OA.
2. Construct a circle with
center M and radius MA.
Label the points where
the circles intersect as
3. Draw AB and AC.
Band C.
I
I
~
A
I
A
XI
You can show thatAB
=AC in Construction 14.
THEOREM 13.5: The two segments tangent to a circle from a
point outside the circle are congruent.
1. When will a segment or ray be tangent to a circle?
2. Look at Construction 13. Why is AB tangent to 00?
3. OB and OC have been added to the completed figure
for Construction 14.
a. LOCA and LOBA are inscribed angles of OM.
Narne the intercepted arc for each angle.
b. mLOCA
= :.I and mLOBA =
c. Why are AB and AC tangent to 00?
13.3 Tangents
389
Class Exercises
Draw circles similar to those shown. Draw all common tangents for
each pair of circles.
·oo ·co J.O
Ao
1
2
Draw a figure similar to, but larger than, the one shown.
4. Construct the line tangent to OC at point A.
5. Construct two lines through B that are tangent to OC.
•B
Exercises
1. Draw a circle and select any point outside the circle. Construct two
lines through this point that are tangent to the circle.
2. Draw a circle and select any point on the circle. Construct the line
tangent to the circle at this point.
CX and CY are tangent to OD at points X and Y.
3.
4.
5.
6.
mLDXC = · • and mLDYC =
If DX= 8, thenDY=
If CX = 20, then CY =
Name two isosceles triangles.
Draw circles similar to those shown. Draw all common tangents for
each pair of circles.
7.
cO 8.00 9.0 0 Q) 11.0
10.
Find each length. 0 and P are centers of circles. Tangents are shown in
Exercises 13-17.
12. OP
13. a. PB b. PD
14. a. GO b. FE
B
390
Chapter 13
Circles
12
!
15. a. AB
16. a. RS
b. Area of 6ABO
b. ST
c. RT
d. Perimeter of 6 RST
e. mLS (Why?)
B
17. a. AW
b. AD
c. YD
d. XC
e. Perimeter of ABCD
f. Area of ABCD
f. Area of 6 RST
'
,A
z
R
4
18. a.
~T
C ~ _ ...
'D
12
PA and PB are tangents to OC. N arne a supplement
of LAPB. Justify your answer.
b. 6PDA =: . by .
c. mLADP =
d. Why is 6PDA - 6ADC?
e. N arne four complements of L CAD.
APPLICATION
19. Distance to the Horizon The diagram at the right represents a
cross section of Earth. The radius of Earth is about 6,400 km. Use a
calculator to find the distance d that a person could see on a clear
day from each of the following heights h above Earth. Round your
answer to the nearest tenth of a kilometer.
a. 100m
b. 200m
c. 300m
Seeing in Geometry
A nickel, a dime, and a quarter are tangent as
shown. Tangents are drawn to the points of
tangency from point A. If AB = 1 in., how
long is AE? Give a reason for your answer.
A
13.3 Tangents
391
Thinking About Proof
Using Definitions
The following statement, which you can use as a reason in proofs,
follows directly from the definitions of a circle, congruent circles, and
congruent segments.
Radii of a circle (or congruent circles) are congruent.
Exercise
To be Proven: The two segments tangent to a circle from a point
outside the circle are congruent (Theorem 13.5).
Given:
Prove:
AB and AC are tangent to 00.
- AB := AC
B
8>A
c
Note: To prove Theorem 13.5, you can add
OA and radii OB and OCto the diagram.
B
~A
c
The statements and corresponding reasons for a proof of Theorem
13.5 are given below in a scrambled order. Put the statements on the
left in a logical order and match each with an appropriate reason
chosen from the list on the right.
(1) AB and AC are tangent to 00.
(a) Definition of a right angle
- (2) AB := AC
(b) Definition of a right triangle
(3) 60BA and 60CA are right triangles.
(c) A segment is congruent to itself.
(4) LOBA and LOCA are right angles.
(d) Given
(5) 60BA := 60CA
(e) CPCTC
(6) AB
-
_l
OB and AC
_l
-
OC
(f) HL Theorem
(7) OB := OC
(g) Radii of a circle are congruent.
(8) OA := OA
(h) If a line is tangent to a circle, then the
line is perpendicular to the radius
drawn to the point of tangency.
392
Chapter 13
Circles
t
13.4
Chords and Arcs
Objective: To apply theorems about chords of circles.
The following Exploring activity uses the fact that all radii of a circle are
congruent.
EXPLORING
Part A
In OO,AB
=CD. Give a reason for each statement.
1. MOB:=~DOC
2. LAOB := LDOC
3. AB :=DC
PartB
In OX, EF
=Gil. Give a reason for each statement.
1. LEXF := LHXG
2. ~EXF := ~HXG
3. EF:=HG
Parte
In OP, PD .l AB. Give a reason for each statement.
1.
~APC
:=
~BPC
~
2. AC := BC
.....
3. LAPC := LBPC
4. AD:= BD
A
B
D
The EXPLORING activity demonstrates the following theorems.
THEOREM 1 3.6: In a circle (or congruent circles), congruent
chords determine congruent arcs and congruent arcs determine
congruent chords.
THEOREM 13.7: In a circle, a diameter that is perpendicular to
a chord bisects the chord and its arc.
13.4 Chords and Arcs
393
Example 1: Given: AB := CD; PX ..L AB and PY ..L CD
a. Name three segments congruent to XB.
B
b. Why is 6. PXB := 6. PYC?
c
c. Why is PX := PY?
Solution: a. AX, DY, and YC
-
-
-
b.XB := YC and PB := PC
LPXB and LPYC are right angles.
Thus 6. PXB := 6. PYC by HL.
c. CPCTC
Exam pie 2: Given: PX := PY; PX ..L AB and PY ..L CD
a. Why is 6.PXB := L::.PYC?
b. Why is XB =: YC?
c. Why is AB =: DC?
B
Solution: a. HL
b.CPCTC
c. Since PX and PY are perpendicular to AB and CD,
respectively, they bisect AB and CD. Since XB
YC,
- AB =:CD.
=
The Examples demonstrate the following theorem.
THEOREM 13.8: In a circle, congruent chords are equidistant
from the center and chords which are equidistant from the center are
congruent.
1. If the radius of OP is 5, AB = 8, and CD
you conclude about PX and PY?
= 6, what can
2. If AB > CD, what can you conclude about PX and PY?
Justify your answer.
3. If PX < PY, what can you conclude about AB and CD?
394
Chapter 13
Circles
B
Class Exercises
Give a reason for each statement about Oc.
2. DA := EA
1. DF := EF
4. LADB is a right angle.
3. DA := EA
D
5. If EF = 6, then DE=
6. If mDE
= 140°, then mAD =
7. If mLABD = 36°, then mAD = [ . and mDE
8. If BC = 5 and BD = 8, then AD =
=• .
Exercises
1. ForOC, if CA
=
=[
9, then CB
2. If AB = 16, then AD =
3~ If mLACS = 64°, then mAS
=l
and RS
~ and
= ·] .
mAE
= • .
AL
4. If mAS = 60°, then mAR =
•
= 130°, then mSB = and mLBCS = • .
= 13 and CD = 5, then AB =
= 30 and CD= 8, thenAC =
fJ
5. If mAE
6. If AC
7. lfAB
1'1.)')
KL and MN are diameters ofOO. Name the following.
8. the midpoint of PT
9. an arc congruent to PK
11. an arc congruent to fJVJ
13. a segment congruent to MN
10. an arc congruent to LT
12. a segment congruent to TR
Give a reason for each statement about OE.
14. If EF := EG, then AB := CD.
15. If AB := CD, then AB := CD.
16. If AB := CD, then EF := EG.
B
20.
B
K
M
®
N
L
T
®
B
D
c
CH =DB
18. AF =BF
For 00 can you conclude that AS =Be? Explain.
17.
19.
p
H
21.A~
0
22.
A
13.4 Chords and Arcs
395
For Oo find each length or angle measure.
23. AO = AB = 8
a. AD
b. OD
b. OE
d.GE
24. a. OD
c. OG
e. FG
25. a. AB
b. AC
c. mLOAC d. mLAOC
e. OC
A
B
APPLICATIONS
Algebra For 00 find x andy.
26.
140°
27.
40°
Test Yourself
@
0
For 00 find each measure or length. In Exercise 3, PR and PS are tangents.
1. a. mAB
b. mLBOC
c. mLBAC
2. a.mFG
woo
A
120
b. mEG
c. XG
®XD
~
12
0
G
5. an obtuse angle inscribed in a circle
For each exercise draw a figure similar to the one shown.
6. Construct a line tangent to OC at point A.
7. Construct two lines tangent to 0 C from point B.
Chapter 13
Circles
R
P
E
Draw a figure to fit each description.
4. all common tangents for two circles like those shown
396
6
b.PO
o
C
o
3. a.PS
65o
F
S
00
Q
•B
13.5
Inscribed and Circumscribed
Polygons and Circles
Objective: To draw and construct inscribed and circumscribed polygons
and circles.
If every vertex of a polygon is on the same circle, we say that the circle is
circumscribed about the polygon and the polygon is inscribed in the circle.
If every side of a polygon is tangent to the same circle, we say that the
polygon is circumscribed about the circle and the circle is inscribed in the
polygon.
Example: OM is circumscribed about
ABCD.
ABCD is inscribed in OM
EFGH is circumscribed about OP
OP is inscribed in EFGH.
A~c
H
PE
Constructions 15 and 16 are based on the following theorems, which were
stated on page 139:
The perpendicular bisectors of the three
sides of a triangle intersect in a point that is
equidistant from the vertices of the triangle.
The three angle bisectors of a triangle
intersect in a point that is equidistant from
the sides of the triangle.
- - - - - - - - · CONSTRUCTION 1 5
Construct a circle circumscribed about ~ABC.
1. Construct the perpendicular bisectors
of any two sides of the triangle. Label
the point of intersection as 0.
2. Construct a circle using 0 as the center
and either OA, OB, or OC as the radius.
c
A £..._____,..~~---l.
A
I
13.5 Inscribed and Circumscribed Polygons and Circles
397
CONSTRUCTION 16
Construct a circle inscribed in ~ABC.
1. Construct the bisectors of any two
angles of the triangle. Label the
point of intersection as 0.
c
2. Construct a perpendicular from 0 to any
side of the triangle. Label the point of
intersection as X. Construct a circle using
0 as the center and OX as the radius.
c
A
B
A
You can use either a compass or a protractor to inscribe certain polygons in
a given circle.
EXPLORING .
Part A
1. Use a compass to construct a circle. Using the same compass
setting, construct arcs to mark off six points on the circle.
2. Use a straightedge to connect the points, forming the polygon shown.
3. Draw dashed lines to connect the vertices of the polygon to the
center of the circle. What kind of triangles are formed? Why?
4. What is the measure of each angle of each triangle? Why?
5. Describe the inscribed polygon.
PartB
1. If ABCDE is a regular pentagon inscribed in Oo, what is the
measure of each central angle shown?
2. Construct a 00. Use a protractor and straightedge to inscribe a
regular pentagon in Oo.
3. How could you use a protractor and straightedge to inscribe a
regular octagon in a circle?
398
Chapter 13
Circles
A
1. How could you modify Part A of the Exploring activity to construct an
inscribed equilateral triangle?
2. In the figure at the right, diameters AB and CD are perpendicular.
a. Why is ACBD a square?
A
D~C
b. Describe how you could modify this figure to construct an
B
inscribed regular octagon.
3. Describe how to construct a circle passing through any three
noncollinear points.
4. a. Must the perpendicular bisector of a chord of a circle contain the
center of the circle? Why or why not?
b. Suppose you were given an arc of a circle, such as the one at the
right. How could you use two chords to find the center of
the circle?
Class Exercises
1. Which of the four figures show(s) a
circle circumscribed about a triangle?
Which show(s) a circle inscribed in
a triangle?
r
A©B@<qDQ
2. Draw a large triangle. Use Construction 15 to
construct a circle circumscribed about the triangle.
3. Draw a large triangle. Use Construction 16 to construct
a circle inscribed in the triangle.
4. Use a protractor to draw an inscribed regular 9-sided polygon.
Exercises
Name the figures (A-F) that fit each
description.
1. a triangle inscribed in a circle
2. a triangle circumscribed about a circle
3. a circle inscribed in a triangle
AQB®C~
Dc:J E@ ~
4. a circle circumscribed about a triangle
13.5 Inscribed and Circumscribed Polygons and Circles
399
For each exercise begin with a circle. Use a protractor and straightedge
to draw each inscribed polygon described.
5. a regular hexagon
6. an equilateral triangle
8. a regular octagon
7. a square
Construct each of the following.
9. a circle circumscribed about an obtuse triangle
10. a circle circumscribed about a right triangle
11. a circle inscribed in an acute triangle
12. a circle inscribed in an obtuse triangle
Draw a figure to fit each description.
13. a right triapgle inscribed in a circle
14. an obtuse triangle inscribed in a circle
15. an obtuse triangle circumscribed about a circle
16. a rectangle inscribed in a circle
17. a trapezoid circumscribed about a circle
18. a rectangle circumscribed about a circle
For each exercise begin with a circle. Use a compass and straightedge to
construct each inscribed polygon described.
19. an equilateral triangle
20. a square
21. a regular octagon
22. a regular dodecagon
23. Find each of the following.
a.AC
b.mLABC
c. AD
d. area of ABCD
e. area ofOO
f. area of shaded region
24. Draw a figure like the one shown at the right.
Locate the center of the circle.
25. Construct an equilateral triangle circumscribed
about a circle. (Hint: Locate the vertices for an
inscribed equilateral triangle. Construct a tangent
to the radius drawn to each point.)
400
Chapter 13
Circles
0
26. Construct a rectangle circumscribed about a circle.
APPLICATIONS
27. Archaeology An archaeologist found part of a circular plate, as
shown at the right. Draw a similar figure, locate the center of the
circle, and complete the circle.
9
28. Design Use a compass to construct the design shown.
(Hint: Construct a circle. With the compass point on the
circle and the same radius, construct an arc. Use the
endpoints of that arc as centers for the next arcs.)
29. Computer The following LOGO procedures use the CIRCLE
procedure on page 284. Edit each procedure so that it draws a square
inscribed in a circle.
a. To inscr.sq 1 :r
circle :r
pu fd :r pd
rt 90
repeat 4[fd :r * sqrt 2 rt 90]
pu home pd
end
b. To inscr.sq2 :r
circle :r
pu lt 45 fd :r pd
rt 135
repeat 4[fd :r rt 90]
pu home pd
end
Computer
Write LOGO procedures to draw each of the following. You can use the
CIRCLE procedure on page 284.
1. a regular hexagon inscribed in a circle
2. a square circumscribed about a circle
3. an equilateral triangle inscribed in a circle
13.5 Inscribed and Circumscribed Polygons and Circles
401
1 c3 .'6
Angles Formed by Chords,
Tangents, and Secants
Objective: To determine the measures of angles formed by chords,
tangents, and secants.
A secant of a circle is a line that contains a chord. In the
figure at the right, AB and CD are secants. Notice that a secant
intersects a circle in exactly two points.
In the Exploring activities you will investigate the relationship
between certain angles and the arcs they intercept.
EXPLORING
Part A
1. In the figures below, notice that as point C moves closer to point B,
becomes more like a tangent. What do you think is the relationship
between mL1 and mAC in the last figure, where Band C coincide?
BC
A
B
B
B
B=C
2. Begin with three or four circles. In each circle draw an angle formed by
a chord and a tangent. Measure each angle and its intercepted arc. How
do the measures compare?
PartB
1. Narne the arcs intercepted by L 1 and its vertical angle.
2. Is L 1 a central angle of the circle? What is the
relationship of mL 1, mL2, and mL 3? Why?
3. What are mL2 and mL3? Why? What is mL1?
4. What is the relationship between mL 1, mBC, and mAi5?
402
Chapter 13
Circles
A
30°
D
80°
This EXPLORING activity demonstrates the following theorems.
THEOREM 13.9: The measure of an angle formed by a tangent
and a chord is half the measure of the intercepted arc.
THEOREM 13.10: The measure of an angle formed by two
chords that intersect inside a circle is half the sum of the measures
of the intercepted arcs.
EXPLORING .
155°
1. Name the arcs intercepted by LA.
2. Find mL1 and mL2.
A"""
3. Find mLA.
c
4. Repeat Steps 1-3 for 00. AB is a tangent.
(Hint: Apply one of the theorems above.)
5. Repeat Steps 1-3 for OP.
AB and AC are tangents.
A
A
D
6. What is the relationship in each case between mLA and the measures of
the intercepted arcs?
This EXPLORING activity demonstrates the following theorem.
THEOREM 13.11: The measure of an angle formed by two secants,
a tangent and a secant, or two tangents drawn from a point outside the
circle is half the difference of the measures of the intercepted arcs.
13.6 Angles Formed by Chords, Tangents, and Secants
403
Example:
Solution:
Find the measure of each numbered angle.
mL1 = ~ (60° + 100°)
= .!. (160°) = 80°
2
mL2 = .!.2 (1 00° - 40°)
= .!.2 (60°) = 30°
mL3 = .!.2 (100° + 90°)
= .!.2 (190°) = 95°
1. Describe how Theorem 13.3 on page 388 can be considered to be a
special case of Theorem 13.9.
2. Theorem 13.11 deals with the angle formed by two secants that
intersect outside a circle.
a. What theorem deals with the angle(s) formed by two secants that
intersect inside a circle? (Hint: Recall the definition of a secant.)
b. What theorem deals with the angle formed by two secants that
intersect on a circle?
Class Exercises
Find the measure of each arc or angle. In Exercise 1, BE is a tangent.
b. LBAC
2. a.L1
c. DA
d. LDAE
e. LCAE
c.L3
1. a. LCAD
3. a.L1
b. L2
d.
RW
b.L2
s
goo
goo
404
Chapter 13
Circles
Exercises
Find the measure of each arc or angle. When point 0 is shown, it is the
center of the circle. You may assume that segments that look like
tangents are tangents.
1. a. Ll
b.L2
2. Ll
3. a. AB
b. Ll
120°
80°
30°
240°
140°
4. Ll
5. a. Ll
b.AB
6. a.
AcE
b. Ll
A
210°
B
7. a. Ll
b. L2
8. a. Ll
9.a. Be
b. LBAC
c. LEAB d. LEAC
b. L2
160°
E
A
150°
140°
10. a.
c.
Ai5
AEi3
b.LF
11. a. LR
c. LRZS
b. LRTY
12. a.
d. LTSZ
Be
b. LC
c. LCEB
e. LTXZ
B
70°
F
100°
R
A
90°
13.6 Angles Formed by Chords, Tangents, and Secants
405
Find the measure of each arc or angle. In Exercise 14, 0 is the center of
the circle. Segments and rays that look like tangents are tangents.
b. Al5
14. a. J5E
b. F£
c. l5F
13. a. Bi5
d.f5EF
e. LB
f. LA
B
50°
B
F
E
A
APPLICATIONS
Algebra
Find x.
15.
16.
~ 4) 0
17.
(4x
(Sx
+ 4)
+ 2)
0
0
Everyday Geometry
A communications satelliteS is in orbit 22,000 mi above Earth's equator.
The circle shown at the right represents the equator. A and B are the farthest
points on the equator that are in a direct line of communication with the
satellite. mLS = 18°.
1. Find mLSAB.
2. Find mAB.
3. The diameter of Earth is about 8,000 mi. Find the distance on the
surface of Earth between A and B. Use 3.14 for 1t.
406
Chapter 13
Circles
s
Segments of Chords; Secants,
and Tangents
13.7
Objective: To determine the lengths of segments of chords, secants, and
tangents.
You may find the result developed in the following Exploring activity a
little surprising.
EXPLORING .
Give a reason for each statement.
1. mLA = ~ mBC and mL D = ~mBC
2. LA := LD
3. LAXC := LDXB
4. 6AXC - 6DXB
ex
AX
5. DX
=
B
c
BX
6. AX • BX = CX • DX
7. If BX = 15, CX = 18, and DX = 10,
use the equation in Step 6 to find AX.
The EXPLORING activity demonstrates the following theorem.
THEOREM 13.12: If two chords of a circle intersect, 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.
The following terms are illustrated in the figure at the right.
A
PAis a tangent segment.
p
PC is a secant segment.
PB is an external secant segment.
13.7 Segments of Chords, Secants, and Tangents
407
You will justify the following theorems in the Thinking Critically questions.
THEOREM 1 3.1 3:
If two secants intersect in the exterior of a
circle, then the product of the lengths of one secant segment and its
external segment is equal to the product of the lengths of the other
secant segment and its external segment.
THEOREM 1 3.14:
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.
The following diagrams illustrate the theorems. Notice that e, g, and s are
lengths of external secant segments.
Theorem 13.12
Theorem 13.13
f
a•b = c•d
f•e = h•g
Example:
Solution:
r •s =
Find the value of x.
a.
b.
a. 21 • x = (8 + 7)7
b.
21x =
X
=
X=
408
Theorem 13.14
Chapter 13
Circles
15 • 7
105
2T
5
18 • X = 92
18x = 81
X= I!_
18
X =
4l2
t2
1. a. Name two angles whose measures equal ~ mQT.
b. N arne two pairs of congruent angles for f::. PRT and
f::.PSQ.
c. Why is f::. PRT - f::. PSQ?
d. Complete and give a reason:
p
~ = PT
e. Complete and give a reason: PR • • = PT • •
f. What have you just shown to be true?
2. a. Write two congruence statements between angles of
6. PQS and 6. PTQ. Explain why each congruence is
true.
b. Write a similarity statement between the triangles.
Why is it true?
p
c. Write a proportion that involves PQ and PS.
d. Apply the means-extremes property to your
proportion in part (c). What have you just shown to
be true?
Class Exercises
Find the value of x.
1.
--
2.
3.
4.
-Exercises
Find the value of x.
1.
2.
3.
4.
13.7 Segments of Chords, Secants, and Tangents
409
7.e:?
Find the value of x.
5.
6.
~
~
9
X
lO.EEJ
9.~
X
8.
11.
X
6
12.
13.
~
APPLICATION
14. Estimation You can estimate the diameter of a circular
swimming pool by holding the ends of a yardstick against
the inside wall of the pool and measuring the distance
from the midpoint of the yardstick to the wall. To the
nearest foot, what diameter of the pool corresponds to
each of the following distances from the midpoint to the
wall? (Hint: Use Theorem 13.12.)
a. 1 in.
7 .
b • 8 1n.
1 .
c. 18 1n .
18 in.
18 in.
~~?~
d. 1~ in.
Everyday Geometry
A train wheel consists of the main wheel, which
rides on the rail, and behind it a larger flange
wheel, which holds the train on the track. In the
figure at the right, MN = 1 in. and AB = 12 in.
Find the radius of each wheel.
main
flange
wheel
wheel
rail
410
Chapter 13
Circles
13.8
-,
...
Locus of Points
Objective: To describe the set of points that satisfy a given set of
conditions.
A locus is the set of all points (and only those points) that satisfy a given set
of conditions. For example, the locus of points in a plane that are 5 em from
a given point 0 in the plane is the circle with center 0 and radius 5 em.
Without the restriction "in a plane" the locus would be the sphere with
center 0 and radius 5 em. In this lesson you may assume, unless otherwise
stated, that all solutions are in a plane.
EXPLORING
1. Describe the locus of points equidistant from two points A and B.
2. Describe the locus of points equidistant from the sides of LCDE.
v~----------+-•
E
Drawing a diagram will often help you solve a locus problem. Sometimes a
diagram helps you to describe a locus.
Example: Describe the locus of points equidistant from two parallel
lines, l and m.
Solution: Find three or four points that satisfy the conditions.
~--------------------------------------~~ [
• • • •
~-------------------------------------~ m
Think of the location of other points satisfying the
conditions. Draw the figure formed by these points and
describe the locus.
~-----------------------------------------~ [
~ -------------- · - · - · - · -------------- ~
~---------------------------------------------~ m
The locus is the line halfway between l and m and parallel to
both lines.
13.8 Locus of Points
411
1. a. For which types of quadrilaterals (parallelogram, rectangle,
rhombus, square, trapezoid) will a diagonal always follow the
same path as the locus of points equidistant from any two
adjacent sides?
b. Describe the locus of points equidistant from two adjacent sides
of a square.
c. Describe the locus of points equidistant from sides
AB and AD of rhombus ABCD. Use a diagram to
help you describe the locus.
d. Describe the locus of points equidistant from sides
AB and BC of rhombus ABCD. Use a diagram to
help you describe the locus.
,--------,C
A....____ _ _ _
For Questions 2 and 3 do not limit your answers to a plane.
2. What is the locus of points equidistant from two parallel planes?
3. What is the locus of points equidistant from two points?
Class Exercises
If AB = 6 em, describe each locus.
1. the locus of points that are 3 em from A and 3 em from B
2. the locus of points that are 3 em from A and 4 em from B
3. the locus of points that are 3 em from A and 2 em from B
Exercises
Describe each locus.
1. the locus of points 3 em from a given line l
2. the locus of points equidistant from two perpendicular lines
3. the locus of midpoints of all radii of a circle that has center 0
and radius 4 em
4. the locus of points equidistant from all four vertices of a
rectangle
5. the locus of points 3 em from a given segment, AB
412
Chapter 13
Circles
___J
6. the locus of points equidistant from sides AD and BC of rectangle ABCD
7. the locus of the centers of all circles tangent to a line lata given point P on l
8. the locus of points equidistant
A
from AB and AC, where AB and AC are
legs of isosceles h. ABC.
BDC
9. R and S are any two points. Find the locus of points that are 3 em from
Rand 4 em from S. (Hint: Consider three cases.)
10. X, Y, and Z are any three noncollinear points. Find the locus of points
that are 2 em from X and equidistant from Y and Z.
11. Describe the locus of the center of a wheel with diameter
15 in. as it rolls along a straight path on a flat surface.
(Note that this problem is not restricted to a plane.)
APPLICATION
~7
12. Distance A pizza company advertises free delivery within a 5-mi
radius of its store. That is, free deliveries are made to any location
inside or on a circle with the pizza store as center and a radius of 5 mi.
Will the distance traveled round trip on a single delivery always be
10 mi or less? Why or why not?
Thinking in Geometry
The tree on the treasure map is 30 paces
from the river. The instructions for finding
the buried treasure say that it can be found
40 paces from the river and 15 paces from
the tree. In how many locations might you
have to dig in order to find the treasure?
Give a reason for your answer.
N
w-$-E
s
;{i"
:;;~·;.~\
~>,.(~~
~tree
t
30 paces
13.8 Locus of Points
413
13.9
Problem Solving Application:
Circle Graphs
Objective: To display data using a circle graph.
Sometimes circle graphs are used to display information.
Exam pie: The table shows the results of a poll of
skiers at the Ponderosa Mountain Ski
Resort. Display the results in a circle graph.
Ponderosa Skiers by Age
Age
Percent of all
skiers
0-10
10%
Solution: 1. Write each percent as a
decimal and multiply by 360°
to find the measure of the central
angle. Round to the nearest degree.
0-10
11-20
21-30
31-40
over 40
0.1
0.15
0.25
0.32
0.18
X
X
X
X
X
360°
360°
360°
360°
360°
= 36°
= 54°
= 90°
= 115.2° :::: 115°
= 64.8° :::: 65°
2. Draw a circle. Use a protractor
to draw each central angle.
11-20
15%
21-30
25%
31-40
32%
18%
Over40
3. Label each section and write a title
for the graph.
Ponderosa Skiers by Age
Notice that the relative sizes of the categories can be gauged simply by
comparing the central angles in a circle graph. The smallest category is
clearly the "0-10" category. The largest is "31-40."
Other relationships can be seen by comparing groups of categories. Notice
that half of all skiers come from the "0- 30" age group. One quarter are in
the "0-20" age group.
Class Exercises
Animals' Continents of Origin
Answer these questions about the continents of
origin of the animals at Central Zoo.
1. Which continent has the largest
representation at the zoo? Which has the
smallest representation?
2. Which two continents together with South
America produced half of the zoo population?
3. Find the measure of each central angle.
a. North America (25%)
b.Europe (8%)
c. Oceania (2%)
Exercises
Methods of commuting
Answer these questions about commuting
methods in one metropolitan area.
1. What is the most popular method of commuting?
2. What methods together with riding the subway
represent half of all commuting methods?
3. Find the measure of each central angle.
a. subway (35%)
b. car (42%)
c. bus (8%)
Make a circle graph for the given data.
4. Preference for Mayor
Candidate
Percent
5. Elements in Earth's Crust
Element
Percent
6. Family Budget
Category
Percent
Smith
55%
Oxygen
48%
Rent
30%
Jones
30%
Silicon
27%
Food
25%
Green
10%
Aluminum
8%
Utilities
12%
Other
5%
Other
17%
Clothing
10%
Transportation 10%
Miscellaneous 13%
13.9 Problem Solving Application: Circle Graphs
415
Chapter 13 Review
Vocabulary and Symbols
You should be able to write a brief statement, draw a picture, or give an
example to illustrate the meaning of each term or symbol.
Vocabulary
arcs
congruent(p. 382)
length (p. 381)
major (p. 381)
measure of (p. 381)
minor (p. 381)
semicircle (p. 381)
circles
circumscribed (p. 397)
congruent(p. 382)
inscribed angle (p. 385)
locus (p. 411)
point of tangency (p.388)
polygons
circumscribed (p. 397)
inscribed (p. 397)
Symbols
secant (p. 402)
AB
segments
external secant (p. 407)
PQR
tangent
common (p. 388)
internal (p. 388)
external (p. 388)
line (p. 388)
minor arc AB (p. 381)
major arc PQR or
semicircle PQR (p. 381)
secant (p. 407)
tangent (p. 407)
inscribed (p. 397)
Summary
The following list indicates the major skills, facts, and results you should
have mastered in this chapter.
13.1 Determine the measures of arcs of circles. (pp. 381-384)
13.2 Use the relationship between the measures of an inscribed angle and
its intercepted arc. (pp. 385-387)
13.3 Construct tangents and apply theorems about tangents. (pp. 388-391)
13.4 Apply theorems about chords of circles. (pp. 393-396)
1 3. 5 Draw and construct inscribed and circumscribed polygons and
circles. (pp. 397- 401)
13.6 Determine the measures of angles formed by chords, tangents, and
secants. (pp. 402-406)
13.7 Determine the lengths of segments of chords, secants, and tangents.
(pp. 407-410)
13.8 Describe the set of points that satisfy a given set of conditions.(pp. 411-413)
13.9 Display datct using a circle graph (pp. 414-415)
416
Chapter 13
Review
Exercises
Find each measure for OC. BF is a tangent.
1.mLACD
2.mADB
5. mLEBF
4. mBE
60°
3.mLDBE
6. mLDBA
Find each measure or length. Tangents are shown in Exercises 7 and 10.
7. STand SO
8. mL1
9. z
I
If
;:;,
n
\jj1
70° ~ 100°
S
T
11.
X
InOO, OX= 17 and OC =OZ =8.
13. Find mXY if mLXOZ = 62°.
14. FindXY.
15. Name two congruent chords and two congruent arcs.
12./e
®
X
z
B
y
For each exercise, begin with OP or L. ABC.
16. Draw Q on OP and construct the tangent to OP at Q.
17. Draw X not on OP and construct two tangents to OP from X.
18. Construct a circle circumscribed about MBC.
19. Inscribe a square in OP.
20. Given diameter JK ofOO, describe the locus of points L such that
LJLK is a right angle.
PROBLEM SOLVING
21. Make a circle graph to show the distribution of the world's population:
North America, 7%; Latin America, 10%; Europe, 12%; U.S.S.R., 7%; Asia,
48%; Africa, 15%; Oceania (some of the islands in the Pacific Ocean), 1%.
Chapter 13 Review
417
Q
o
Chapter 1 3 Test
£j
Find each measure or length. In Exercises 1, 5, and 6, 0 is the center of
the circle. Tangents are shown in Exercises 2, 6, and 8.
1. mAB and mL1
2. mLDEF and mL2
3. x
D
120°
4. y
5. MP
6. OB and CB
J
12
1.y
8.
9. mL3 and mL4
X
80°
10. Draw an acute L XYZ and inscribe a circle in L XYZ.
11. Draw an obtuse
L
PQR and circumscribe a circle about
L
PQR.
12. Draw a circle and inscribe a regular hexagon in the circle.
13. Draw OJ and point Z on OJ. Construct the tangent to OJ at Z.
14. Describe the locus of points equidistant from the vertices of a triangle.
PROBLEM SOLVING
15. Make a circle graph to show the distribution of the U.S. population by
age: under age 5, 7%; ages 5-19, 25%; ages 20-44, 37%; ages 45-62,
20%; over age 62, 11%.
418
Chapter 13
Test
B
100°