Unit 13 - Circle Workbook 13 - Circle... · The diameters of (L and (M are 20 and 13 units,...

Geometry

Circles

Name _________________

Period _____________

1

!

"

Date: ______________________

Notes

Section 10 – 1 & 10 – 2: Circumference, Arcs, and Angles

Circle – a set of _________ equidistant from a given point called the __________ of the

circle

Circumference:

Example #1:

a.) Find C if r = 13 inches. b.) Find C if d = 6 millimeters.

c.) Find d and r to the nearest hundredth if C = 65.4 feet.

Central Angle – an angle that has the center of the circle as its vertex, and its sides

contain two radii of the circle

Minor Arc – arc degree measure equals the measure of the ____________ angle and is

less than 180 degrees

Major Arc – arc degree measure equals 360 minus the measure of the ___________ arc

and is greater than 180 degrees

2

#

Semicircle – arc degree measure equals __________

Example #1: Refer to circle T.

a.) Find .m RTS!

b.) Find .m QTR!

Example #2: In circle P, 46m NPM! " , PL bisects ,KPM! and .OP KN# Find each

measure.

a.) m OK

b.) m LM

c.) m JKO

3

© Glencoe/McGraw-Hill Glencoe Geometry

For Exercises 125, refer to the circle.

1. Name the circle. 2. Name a radius.

3. Name a chord. 4. Name a diameter.

5. Name a radius not drawn as part of a diameter.

6. Suppose the radius of the circle is 3.5 yards. Find the diameter.

7. If RT 5 19 meters, find LW.

The diameters of (L and (M are 20 and 13 units, respectively.

Find each measure if QR 5 4.

8. LQ 9. RM

The radius, diameter, or circumference of a circle is given. Find the missing

measures to the nearest hundredth.

10. r 5 7.5 mm 11. C 5 227.6 yd

d 5 , C < d < , r <

Find the exact circumference of each circle.

12. 13.

SUNDIALS For Exercises 14 and 15, use the following information.

Herman purchased a sundial to use as the centerpiece for a garden. The diameter of the

sundial is 9.5 inches.

14. Find the radius of the sundial.

15. Find the circumference of the sundial to the nearest hundredth.

40 mi

42 mi

K24 cm

7 cm

R

PQL R

MS

L

W

R

S

T

Practice

Circles and Circumference

NAME ______________________________________________ DATE ____________ PERIOD _____

10-110-1

4

NAME ___________________________ Period _________ Date _____________

Today you will be INSCRIBING polygons. Follow the directions carefully. Mark all points

clearly and answer questions asked.

Part 1

Draw !O in the area below.

Draw ABC∆ inscribed in !O with BC as a diameter.

What is the m BC _______________________

What is BACm∠ __________

What type of triangle is ABC∆ ? ________________

Is BACm∠ = m BC _____________

How do the measures of BAC∠ and BC compare? _____________________________

Write a conjecture that compares the inscribed angle and the arc it intercepts:

5

Part 2

You will draw a regular pentagon (called PENTA) by following the directions:

• How big is one central angle of a pentagon? (Remember we did total circle divided by

number of sides) _____________

• Draw an angle PKE∠ that is the measure of one central angle using the protractor.

• Using the same vertex point (K), draw an adjacent angle to PKE∠ also the same measure

using the protractor.

• Repeat this for all 5 angles of the Pentagon

• Draw a new circle called !K using the point K of the angles you drew as the center using

the compass.

• Complete the Pentagons by connecting the points (chords) where the circle and the angle

sides intersect. Call these points PENTA.

How big is each central angle of Pentagon PENTA? ___________

How big is each vertex angle of Pentagon PENTA? __________-

What is the measure of each of the congruent arcs? __________

Complete the measures below:

APEm∠ _______ m PE _________ m ENA _________

PENm∠ _______ m EN _________ m NTP __________

ENTm∠ _______ m NT _________ m TAE __________

NTAm∠ _______ m TA _________ m APN __________

TAPm∠ _______ m AP _________ m PET __________

APE∠ forms which arc? _______

APE∠ PEN∠ forms which arc? _______

APE∠ TAP∠ forms which arc? _______

How does the inscribed angle compare to the arc it forms?

6


ALGEBRA In (Q, AwCw and BwDw are diameters. Find each

measure.

1. m/AQE 2. m/DQE

3. m/CQD 4. m/BQC

5. m/CQE 6. m/AQD

In (P, m/GPH 5 38. Find each measure.

7. mEFC

8. mDEC

9. mFGC

10. mDHGC

11. mDFGC

12. mDGEC

The radius of (Z is 13.5 units long. Find the length of each arc

for the given angle measure.

13. QPTC

if m/QZT 5 120 14. QRC

if m/QZR 5 60

15. PQRC

if m/PZR 5 150 16. QPSC

if m/QZS 5 160

HOMEWORK For Exercises 17 and 18, refer to the table,

which shows the number of hours students at Leland

High School say they spend on homework each night.

17. If you were to construct a circle graph of the data, how many

degrees would be allotted to each category?

18. Describe the arcs associated with each category.

Homework

Less than 1 hour 8%

1–2 hours 29%

2–3 hours 58%

3–4 hours 3%

Over 4 hours 2%

Q

Z

T

P

R

S

F

P

D

EG

H

(5x 1 3)8

(6x 1 5)8 (8x 1 1)8

QA

B

C

DE

Practice

Angles and Arcs

NAME ______________________________________________ DATE ____________ PERIOD _____

10-210-2

7


The Four Color Problem

Mapmakers have long believed that only four colors are necessary todistinguish among any number of different countries on a plane map.Countries that meet only at a point may have the same color providedthey do not have an actual border. The conjecture that four colors aresufficient for every conceivable plane map eventually attracted theattention of mathematicians and became known as the “four-colorproblem.” Despite extraordinary efforts over many years to solve theproblem, no definite answer was obtained until the 1980s. Four colorsare indeed sufficient, and the proof was accomplished by makingingenious use of computers.

The following problems will help you appreciate some of thecomplexities of the four-color problem. For these “maps,” assume thateach closed region is a different country.

1. What is the minimum number of colors necessary for each map?

a. b. c.

d. e.

2. Draw some plane maps on separate sheets. Show how each can be colored using four colors. Then determine whether fewer colors would be enough.

Enrichment

NAME ______________________________________________ DATE ____________ PERIOD _____

10-110-1

9


Curves of Constant Width

A circle is called a curve of constant width because no matter how

you turn it, the greatest distance across it is always the same.

However, the circle is not the only figure with this property.

The figure at the right is called a Reuleaux triangle.

1. Use a metric ruler to find the distance from P to

any point on the opposite side.

2. Find the distance from Q to the opposite side.

3. What is the distance from R to the opposite side?

The Reuleaux triangle is made of three arcs. In the example

shown, PQC

has center R, QRC

has center P, and PRC

has

center Q.

4. Trace the Reuleaux triangle above on a piece of paper and

cut it out. Make a square with sides the length you found in

Exercise 1. Show that you can turn the triangle inside the

square while keeping its sides in contact with the sides of

the square.

5. Make a different curve of constant width by starting with the

five points below and following the steps given.

Step 1: Place he point of your compass on

D with opening DA. Make an arc

with endpoints A and B.

Step 2: Make another arc from B to C that

has center E.

Step 3: Continue this process until you

have five arcs drawn.

Some countries use shapes like this for coins. They are useful

because they can be distinguished by touch, yet they will work

in vending machines because of their constant width.

6. Measure the width of the figure you made in Exercise 5. Draw

two parallel lines with the distance between them equal to the

width you found. On a piece of paper, trace the five-sided figure

and cut it out. Show that it will roll between the lines drawn.

A

C

B

D

E

P Q

R

Enrichment

NAME ______________________________________________ DATE ____________ PERIOD _____

10-210-2

10

$

Date: _____________________

Notes

Section 10 – 3: Arcs and Chords

! The ________________ of a chord are also endpoints of an _______.

Arcs and Chords

Theorem 10.2

Ex:

: In a circle, two ___________ arcs are congruent if and only

if their corresponding ____________ are congruent.

! The chords of _______________ arcs can form

Inscribed and Circumscribed

a _______________.

! Quadrilateral ABCD is an ______________ polygon

because all of its _____________ lie on the circle.

! Circle E is ___________________ about the polygon

because it contains all of the vertices of the _______________.

Theorem 10.3

Ex:

: In a circle, if the diameter (or radius) is ________________

to a chord, then it ___________ the chord and its arc.

11

%

Example #1: Circle W has a radius of 10 centimeters. Radius WL is

perpendicular to chord HK , which is 16 centimeters long.

a.) If mHL = 53, find mMK.

b.) Find JL.

Theorem 10.4: In a circle, two ___________ are congruent if and only if

they are __________________ from the center.

Example #2: Chords EF and GH are equidistant from the center. If the

radius of circle P is 15 and 24EF ! , find PR and RH.

12

Name ___________________________________________ Period ______ Date _________

For this exercise you will need a ruler and a piece of string. Use the ruler to draw the straight

lines and measure them. Use the string to measure the arcs and then use the ruler to measure

the string length you found for the arc measure.

In the circle given (� H),

1. Draw a chord AB

2. Draw another chord the same size as AB and call it CD

3. How far is AB from the center H? ________

4. How far is CD from the center H? ________

5. Draw a diameter FG that is ⊥ to AB at a point you call P

6. What is the length of AP ? ________

7. What is the length of BP ? _________

8. Use the string to find the length of AB _________

9. Use the string to find the length of AP _________

10. Use the string to find the length of BP _________

11. Use the string to find the length of CD _________

13


In (E, mHQC

5 48, HI 5 JK, and JR 5 7.5. Find each measure.

1. mHIC

2. mQIC

3. mJKC

4. HI

5. PI 6. JK

The radius of (N is 18, NK 5 9, and mDEC

5 120. Find each

measure.

7. mGEC

8. m/HNE

9. m/HEN 10. HN

The radius of (O 5 32, PQC

> RSC

, and PQ 5 56. Find each

measure.

11. PB 14. BQ

12. OB 16. RS

13. MANDALAS The base figure in a mandala design is a nine-pointed

star. Find the measure of each arc of the circle circumscribed about

the star.

O

QR

P B

S

A

N

ED

X

Y

K

G

H

EK

J

R

I

S

H

Q

P

Practice

Arcs and Chords

NAME ______________________________________________ DATE ____________ PERIOD _____

10-310-3

14


Patterns from Chords

Some beautiful and interesting patterns result if you draw chords toconnect evenly spaced points on a circle. On the circle shown below,24 points have been marked to divide the circle into 24 equal parts.Numbers from 1 to 48 have been placed beside the points. Study thediagram to see exactly how this was done.

1. Use your ruler and pencil to draw chords to connect numbered

points as follows: 1 to 2, 2 to 4, 3 to 6, 4 to 8, and so on. Keep dou-

bling until you have gone all the way around the circle.

What kind of pattern do you get?

2. Copy the original circle, points, and numbers. Try other patterns

for connecting points. For example, you might try tripling the first

number to get the number for the second endpoint of each chord.

Keep special patterns for a possible class display.

37

13

1

25

7 3143 19

44 20

45 21

42 18

41 17

40 16

39 1

5

38 1

4

46 2

2

47 2

3

48 2

4

12 3

6

11 35

10 3

4

9 33

8 32

6 30

5 294 28

3 2

72 2

6

Enrichment

NAME ______________________________________________ DATE ____________ PERIOD _____

10-310-3

15

&

Date: _____________________

Notes

Section 10 – 4: Inscribed Angles

If an angle is ______________ in a circle, then the measure of the angle equals

______________ the measure of its intercepted arc (or the measure of the

__________________ arc is ___________ the measure of the inscribed angle).

Inscribed Angles

Ex:

Example #1: In circle O, mAB = 140, mBC = 100, and mAD = mDC. Find the

measures of the numbered angles.

16

'

Angles of Inscribed Polygons

Theorem 10.7

Ex:

: If an inscribed angle intercepts a semicircle, the angle is a

____________ angle.

Example #2: Triangles TVU and TSU are inscribed in circle P, with SUVU ! . Find the

measure of each numbered angle if 2 9m x" # $ and 4 2 6m x" # $ .

Example #3: Quadrilateral ABCD is inscribed in circle P. If 80m B" # and 40m C" # ,

find m A" and m D" .

17


In (B, mWXC

5 104, mWZC

5 88, and m/ZWY 5 26. Find the

measure of each angle.

1. m/1 2. m/2

3. m/3 4. m/4

5. m/5 6. m/6

ALGEBRA Find the measure of each numbered angle.

7. m/1 5 5x 1 2, m/2 5 2x 2 3 8. m/1 5 4x 2 7, m/2 5 2x 1 11,

m/3 5 7y 2 1, m/4 5 2y 1 10 m/3 5 5y 2 14, m/4 5 3y 1 8

Quadrilateral EFGH is inscribed in (N such that mFGC

5 97,

mGHC

5 117, and mEHGC

5 164. Find each measure.

9. m/E 10. m/F

11. m/G 12. m/H

13. PROBABILITY In (V, point C is randomly located so that it

does not coincide with points R or S. If mRSC

5 140, what is the

probability that m/RCS 5 70?

V

R

S

C

1408

708

NF

E

H

G

RB

A

D

C

1

2

3

4

U

J

G

I

H

1 3

42

B

ZY

XW

1

2

3 4

5

6

Practice

Inscribed Angles

NAME ______________________________________________ DATE ____________ PERIOD _____

10-410-4

18


Formulas for Regular Polygons

Suppose a regular polygon of n sides is inscribed in a circle of radius r. The

figure shows one of the isosceles triangles formed by joining the endpoints of

one side of the polygon to the center C of the circle. In the figure, s is the length

of each side of the regular polygon, and a is the length of the segment from C

perpendicular to AwBw.

Use your knowledge of triangles and trigonometry to solve the

following problems.

1. Find a formula for x in terms of the number of sides n of the polygon.

2. Find a formula for s in terms of the number of n and r. Use trigonometry.

3. Find a formula for a in terms of n and r. Use trigonometry.

4. Find a formula for the perimeter of the regular polygon in terms of n and r.

A

C

a

s

s

2

r r

x° x°

Bs

2

Enrichment

NAME ______________________________________________ DATE ____________ PERIOD _____

10-410-4

19

Date: _____________________

Notes

Section 10 – 5: Tangents

!

Tangents

Tangent

! The point of intersection is called the _________ ___ ____________.

– a line in the plane of a ___________ that intersects the circle in

exactly one ____________.

Ex:

Theorem 10.9

Ex:

: If a line is ______________ to a circle, then it is

______________________ to the ____________ drawn to the point of

_____________.

Example #1: RS is tangent to circle Q at point R. Find y.

20

)

Theorem 10.10

Ex:

: If a _________ is perpendicular to a radius of a circle at its

______________ on the circle, then the line is _____________ to the

circle.

Example #2: Determine whether the given segments are tangent to the given

circles.

a.) BC b.) WE

Theorem 10.11

Ex:

: If two ______________ from the same exterior point are

_____________ to a circle, then they are ________________.

21

*

Example #3: Find x. Assume that segments that appear tangent to circles are

tangent.

Example #4: Triangle HJK is circumscribed about circle G. Find the perimeter of

!HJK if NK = JL + 29.

22


Determine whether each segment is tangent to the given circle.

1. MwPw 2. QwRw

Find x. Assume that segments that appear to be tangent are tangent.

3. 4.

Find the perimeter of each polygon for the given information. Assume that

segments that appear to be tangent are tangent.

5. CD 5 52, CU 5 18, TB 5 12 6. KG 5 32, HG 5 56

CLOCKS For Exercises 7 and 8, use the following

information.

The design shown in the figure is that of a circular clock

face inscribed in a triangular base. AF and FC are equal.

7. Find AB.

8. Find the perimeter of the clock.

F

B

A

D E

C7.5 in.

2 in.

12

6

3

2

48

1011 1

57

9

L

HG

KT

B D

U

V

C

P

T

S10

15

x

L

T

U

S

7x 2 3

5x 1 1

P

R

Q

14

50

48L

M

P

20 21

28

Practice

Tangents

NAME ______________________________________________ DATE ____________ PERIOD _____

10-510-5

23


Tangent Circles

Two circles in the same plane are tangent circles

if they have exactly one point in common. Tangent

circles with no common interior points are externally

tangent. If tangent circles have common interior

points, then they are internally tangent. Three or

more circles are mutually tangent if each pair of

them are tangent.

1. Make sketches to show all possible positions of three mutually tangent circles.

2. Make sketches to show all possible positions of four mutually tangent circles.

3. Make sketches to show all possible positions of five mutually tangent circles.

4. Write a conjecture about the number of possible positions for n mutually tangent circles

if n is a whole number greater than four.

Externally Tangent Circles

Internally Tangent Circles

Enrichment

NAME ______________________________________________ DATE ____________ PERIOD _____

10-510-5

24

Date: _____________________

Notes

Section 10 – 6: Secants, Tangents, and Angle Measures

Secant

Ex:

– a line that intersects a circle in exactly _________ points

Theorem 10.12: (Secant-Secant Angle) Theorem 10.13:

Ex: Ex:

(Secant-Tangent Angle)

Theorem 10.14:

Two Secants Secant-Tangent Two Tangents

Example #1: Find 3!m and 4!m if mFG = 88 and mEH = 76.

25

!!

Example #2: Find m RPS! if mPT = 144 and mTS = 136.

Example #3: Find x.

Example #4: Use the figure to find the measure of the bottom arc.

Example #5: Find x.

26


Find each measure.

1. m/1 2. m/2 3. m/3

Find x. Assume that any segment that appears to be tangent is tangent.

7. 8. 9.

10. 11. 12.

9. RECREATION In a game of kickball, Rickie has to kick the

ball through a semicircular goal to score. If mXZC

5 58 and

the mXYC

5 122, at what angle must Rickie kick the ball

to score? Explain.

goal

B(ball)

X

Z Y

378

x 8

528

x 8

638

x 8

5x 8

628 1168

x 8

598

158

2x 8

398

1018

x 8

2168

3

1348

2

568

1468

1

Practice

Secants, Tangents, and Angle Measures

NAME ______________________________________________ DATE ____________ PERIOD _____

10-610-6

27


Orbiting Bodies

The path of the Earth’s orbit around the sun is elliptical. However, it is often viewed

as circular.

Use the drawing above of the Earth orbiting the sun to name the line or segment

described. Then identify it as a radius, diameter, chord, tangent, or secant of

the orbit.

1. the path of an asteroid

2. the distance between the Earth’s position in July and the Earth’s position

in October

3. the distance between the Earth’s position in December and the Earth’s position

in June

4. the path of a rocket shot toward Saturn

5. the path of a sunbeam

6. If a planet has a moon, the moon circles the planet as the planet circles the sun. To

visualize the path of the moon, cut two circles from a piece of cardboard, one with a

diameter of 4 inches and one with a diameter of 1 inch.

Tape the larger circle firmly to a piece of paper. Poke a pencil

point through the smaller circle, close to the edge. Roll the small

circle around the outside of the large one. The pencil will trace

out the path of a moon circling its planet. This kind of curve is

called an epicycloid. To see the path of the planet around the

sun, poke the pencil through the center of the small circle (the

planet), and roll the small circle around the large one (the sun).

B

A

C

D

J

E

F

G

H

Enrichment

NAME ______________________________________________ DATE ____________ PERIOD _____

10-610-6

28

© Glencoe/McGraw-Hill 0 Glencoe Geometry

Find x to the nearest tenth. Assume that segments that appear to be tangent are

tangent.

1. 2. 3.

4. 5.

6. 7.

8. 9.

10. CONSTRUCTION An arch over an apartment entrance is

3 feet high and 9 feet wide. Find the radius of the circle

containing the arc of the arch.9 ft

3 ft

20

x x 2 6

20

25

x

6

x x 2 3

6

5

15

x

14

1715

x

3

8

10

x

7

21

20

x4

98

x

11 11

5

x

Practice

Special Segments in a Circle

NAME ______________________________________________ DATE ____________ PERIOD _____

10-710-7

29


The Nine-Point Circle

The figure below illustrates a surprising fact about triangles and circles.

Given any n ABC, there is a circle that contains all of the following nine

points:

(1) the midpoints K, L, and M of the sides of n ABC

(2) the points X, Y, and Z, where AwXw, BwYw, and CwZw are the altitudes of n ABC

(3) the points R, S, and T which are the midpoints of the segments AwHw, BwHw,

and CwHw that join the vertices of n ABC to the point H where the lines

containing the altitudes intersect.

1. On a separate sheet of paper, draw an obtuse triangle ABC. Use your

straightedge and compass to construct the circle passing through the

midpoints of the sides. Be careful to make your construction as accurate

as possible. Does your circle contain the other six points described above?

2. In the figure you constructed for Exercise 1, draw RwKw, SwLw, and TwMw. What

do you observe?

A

B

M

S

X

K

T

LY

H O

Z

R

C

Enrichment

NAME ______________________________________________ DATE ____________ PERIOD _____

10-710-7

30


Write an equation for each circle.

1. center at origin, r 5 7 2. center at (0, 0), d 5 18

3. center at (27, 11), r 5 8 4. center at (12, 29), d 5 22

5. center at (26, 24), r 5 Ï5w 6. center at (3, 0), d 5 28

7. a circle with center at (25, 3) and a radius with endpoint (2, 3)

8. a circle whose diameter has endpoints (4, 6) and (22, 6)

Graph each equation.

9. x21 y2

5 4 10. (x 1 3)21 ( y 2 3)2

5 9

11. EARTHQUAKES When an earthquake strikes, it releases seismic waves that travel inconcentric circles from the epicenter of the earthquake. Seismograph stations monitorseismic activity and record the intensity and duration of earthquakes. Suppose a stationdetermines that the epicenter of an earthquake is located about 50 kilometers from thestation. If the station is located at the origin, write an equation for the circle thatrepresents a possible epicenter of the earthquake.

x

y

O

x

y

O

Practice

Equations of Circles

NAME ______________________________________________ DATE ____________ PERIOD _____

10-810-8

31


Equations of Circles and Tangents

Recall that the circle whose radius is r and whose

center has coordinates (h, k) is the graph of

(x 2 h)21 (y 2 k)2

5 r2. You can use this idea and

what you know about circles and tangents to find

an equation of the circle that has a given center

and is tangent to a given line.

Use the following steps to find an equation for the circle that has cen-

ter C(22, 3) and is tangent to the graph y 5 2x 2 3. Refer to the figure.

1. State the slope of the line ø that has equation y 5 2x 2 3.

2. Suppose (C with center C(22, 3) is tangent to line ø at point P. What is

the slope of radius CwPw?

3. Find an equation for the line that contains CwPw.

4. Use your equation from Exercise 3 and the equation y 5 2x 2 3. At what

point do the lines for these equations intersect? What are its coordinates?

5. Find the measure of radius CwPw.

6. Use the coordinate pair C(22, 3) and your answer for Exercise 5 to write

an equation for (C.

Px

y

O

C(22, 3)

y 5 2x 2 3

,

Enrichment

NAME ______________________________________________ DATE ____________ PERIOD _____

10-810-8

32

Geometry vocabulary– circles NAME _____________________________

Period ______ Date _______

Name the following from the drawing with circle W.

B

F

A • W C

G

J D

E

H

_________1. the center of the circle

_________2. a radius

_________3. a diameter

_________4. a chord

_________5. a secant segment

_________6. a tangent segment

_________7. a minor arc

_________8. a major arc

_________9. a semi-circle

_________10. a central angle

_________11. an inscribed angle

_________12. a 90 ° angle

_________13. a point of tangency

33

Geometry

Name: Date:

Instructions: Complete the word search puzzle. Use the clues to help you identify the words.

Copyright © The McGraw-Hill Companies, Inc.

Clues

1. An arc that measures 180. 2. The distance around a circle. 3. In a circle, any segment with endpoints that are the center of the circle and a point on the

circle. 4. In a circle, a chord that passes through the center of the circle. 5. A polygon is _____ in a circle if each of its vertices lie on the circle. 6. A line in the plane of a circle that intersects the circle in exactly one point. 7. An irrational number represented by the ratio of the circumference of a circle to the diameter

of the circle. 8. The locus of all points in a plane equidistant from a given point called the center.

34

Geometry

Copyright © The McGraw-Hill Companies, Inc.

9. _____ arcs − Arcs of the same circle or congruent circles that have the same measure. 10. Any line that intersects a circle in exactly two points. 11. For a given circle, a segment with endpoints that are on the circle. 12. An arc with a measure less than 180. 13. An __________ angle has its vertex on the circle and its sides go through the interior of the

circle 14. A circle is _____ about a polygon if the circle contains all the vertices of the polygon. 15. An arc with a measure greater than 180. 16. A part of a circle that is defined by two endpoints. 17. point of _____ − For a line that intersects a circle in only one point, the point at which they

intersect. 18. _____ angle − An angle that intersects a circle in two points and has its vertex at the center of

the circle.

35

Gra

phic O

rganizer

by

Dale G

raham a

nd

Linda M

eyer

Thom

as C

ount

y

Cent

ral High

Sch

ool

Thom

asville, Ga.

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Unit 13 - Circle Workbook 13 - Circle... · The diameters of (L and (M are 20 and 13 units,...

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Transcript of Unit 13 - Circle Workbook 13 - Circle... · The diameters of (L and (M are 20 and 13 units,...