Name: ___________________________ Period: ________ 10.1 Investigations of Circle theorems

Investigation 1: Angles inscribed in a semi-circle

Step 1: Construct a large circle, and make a diameter with end points A and B

Step 2: Place 1 point somewhere on the arc that connects A and B. Call this point

C.

Step 3: Construct segments 𝐴𝐶 and 𝐵𝐶. Then measure the angle ∠𝐴𝐶𝐵.

Step 4: What is the measurement of ∠𝐴𝐶𝐵?

Step 5: Repeat steps 2 – 4 using new points D and E.

Step 6: Write a summary of your findings that includes your construction, you must include conjecture

that demonstrates your findings.

Investigation 2: Cyclic quadrilaterals (Quadrilaterals whose vertices all lie on a circle)

Step 1: Construct 2 large circles. Place 4 points along the circumference of your

circles and connect these points with segments. Please make sure your shape is not a

rectangle or similar to any shape of your neighbors.

Step 2: Measure each of the angles inside the quadrilaterals. Write the measure of

each angle in its appropriate place.

Step 3: Carefully examine the relationships of the angles of these quadrilaterals.

Step 4: Write a summary of your findings that includes your construction, you must include conjecture

that demonstrates your findings.

Proof 1

Why is the angle inscribed in a semi-circle always a right angle?

1) O is the center of the circle. Draw a segment 𝑂𝐶.

2) What kind of triangle is ΔAOC? _____________________ label the congruent angles “x”

What kind of triangle is ΔOCB? _____________________ label the congruent angles “y”

3) Since the sum of the angles in a triangle is 180°, express ∠𝐴𝑂𝐶 in terms of x and ∠𝐶𝑂𝐵 in terms of y.

∠𝐴𝑂𝐶 = _____________________ ∠𝐶𝑂𝐵 = _____________________

4) What kind of angle pair is ∠𝐴𝑂𝐶 𝑎𝑛𝑑 ∠𝐶𝑂𝐵 ? _____________________

Write an equation relating ∠𝐴𝑂𝐶 𝑎𝑛𝑑 ∠𝐶𝑂𝐵. _______________________________________

5) Use your equations in steps 3 and 4 to algebraically prove that ∠𝐶 is a right angle.

Name______________________________ class ___ 10.2 Constructions Inscribed in Circles

For many constructions, there is more than one way to make the construction. The important thing is

why the constructions work. Challenge yourself to find another way to make these constructions and

be able to prove why your constructions work.

DO EACH CONSTRUCTION AT LEAST 3 TIMES – more if you can find a different way

1.Construct a square inscribed in a circle

a) Use a compass to construct a circle.

b) Construct one diameter

c) Use the compass to create the perpendicular bisector

d) Connect the four outside chords of the circle.

DO EACH CONSTRUCTION AT LEAST 3 TIMES – more if you can find a different way

2. Construct a hexagon inscribed in a circle

a) Construct a circle

b) Construct a radius

c) Set the compass setting to the length of a radius

d) Using that compass setting, create 6 equal distant points around the circle.

3. Construct an equilateral triangle inscribed in a circle

The construction is similar to the hexagon construction in part 2, except you

connect every other vertex to form a triangle.

Challenge: Construct a regular octagon inscribed in a circle.

Name: ___________________________ Period: ________ 10.3 Circle Properties

Solve for the variable. Show your calculations. Mark the diagrams with measurements.

10. Polygon CDFE is a rectangle inscribed in a circle centered at the origin. Find the

coordinates of points D, F and E.

11. The satelite photo below shows only a portion of a lunar crater. How can cartographers use

the photo to find it’s center?

Trace the crater and locate its center. Using the scale shown, find its radius. SHOW YOUR

WORK.

Name: _______________________________Period: ____________ 10.4 Tangent Properties

1. Rays m and n are tangent to 2. Rays r and s are tangent to

Circle P. What is the measure of w? Circle Q, what is the measure of x?

3. Ray k is tangent to circle R. 4. Line t is tangent to both circles. 5. Quadrilateral POST is

What is the measure of y? What is the measure of z? circumscribed about circle Y.

𝑂𝑅 = 13 and 𝑆𝑇 = 12.

What is the perimeter of POST?

6. A satellite in geostationary orbit remains above the same point on the Earth’s surface even as the Earth

turns. If such a satellite has a 30° view of the equator of the earth, what percentage of the equator is

observable from the satellite?

7. Circle P is centered at the origin. 𝐴𝑇 ⃡ is tangent to P at A(8,15). Find the equation of 𝐴𝑇 ⃡ .

8. 𝑇𝐴 and 𝑇𝐵 are tangent to circle O. Something is wrong with this picture. Carefully explain what the

problem is.

Name: ___________________________ Period: ________ 10.5 Apple Pi

3. What are sources of error for this experiment?

4. Which value do you think is greater, the height of the can of tennis

balls or the circumference of the can of tennis balls?

Using r for the radius of a tennis ball, what is the circumference of the

can of tennis balls in terms of r?

Using r for the radius of a tennis ball, what is the height of the can of tennis balls in terms of r?

Which value is greater, the height of the can of tennis balls or the circumference of the can of

tennis balls? Explain your reasoning.

Name: _______________________________Period: ____________ 10.6 Arcs and Angles

Find the indicated measure. SHOW YOUR WORK.

Hint: ↑ what do you know about this line?

8. 9.

10. 11.

Hint: look for inscribed angles

and their arcs

12. Find the measure of each lettered angle. Label the diagram as you go.

a= _____ b = _____ c= _____d= _____e= _____f= _____

g= _____ h= _____ l= _____ m= _____n= _____ p= _____

Name: ______________________________ Period _____________ 10.7 Circumference Problems

1. Alfonzo’s Pizzeria bakes olive pieces in the outer crust of its 25-inch (diameter) pizza. There is at least

one olive piece per inch of crust. How many olive pieces will you get in one slice of pizza? Assume the

pizza is cut into eight slices.

2. To use the machine at right, you turn the crank, which turns the pulley wheel, which

winds the rope and lifts the box. Through how many rotations must you turn the crank

to lift the box 10 feet?

3. A satellite in geostationary orbit stays over the same spot above the planet Jupiter. The satellite

completes one orbit in the same time that Jupiter rotates once about its axis (9 hours and 56 minutes). If

the satellite’s orbit has radius 1.59 × 106 m, calculate the satellite’s orbital speed (In the direction of the

tangent) in meters per second.

4. As you sit in your chair, you are whirling through space with Earth as it moves around the sun. If the

average distance from Earth to the sun is 1.4957 × 1011 m and Earth completes one revolution every

365.25 days, what is your “sitting” speed in space relative to the sun? Give your answer in km/h, rounded

to the nearest 100 km/h.

Thou shouldest remember to use the equations 𝐶 = 2 ∙ 𝜋𝑟 and 𝑠 =𝑑

𝑡.

Thou shalt include thine own calculations. Thou shouldest convert

thy units properly.

Bonus: In problem 8 what would the orbital radius of the beam of light need to be to have the light travel

around the circle exactly 20 times in a second?

5. If the distance from the center of a Ferris wheel to one of the seats is approximately 105 feet, how fast

is a person travelling (in the tangential direction) if the Ferris wheel makes one rotation every 75 seconds?

Express your answer in feet per second to the nearest unit.

6. The diameter of a car tire is about 65 cm. The warranty for the tire is good for 50,000 km. About how

many rotations will the tire make before the warranty expires? (1000m = 1km)

7. While spinning a heavy object on a string that is 38” long a boy twirls the object at 50 revolutions per

minute. After letting the string wrap around his finger several times the object is now flying around 17”

out but at 85 revolutions per minute. Which is travelling faster (tangentially speaking of course).

8. The speed of light in miles per second is: 186,282.397 mi/s. If a beam of light was to circle 15 miles

above the moon’s surface, how many times would it go around the moon in one second? The diameter of

the moon in miles is: 2,159.141 miles.

Name: __________________________ Period: ______ 10.8 Arc Length

1) What is the length of 𝐶�̂�? 2) What is the length of 𝐸�̂�?

3) What is the length of 𝐵𝐼�̂�? 4) If the length of 𝐴�̂� is 6𝜋 m, 5) The radius is 18 feet.

What is the radius? What is the length

of 𝑅�̂�?

6) The radius is 9 m 7) The length of 𝑇�̂� is 12𝜋 8) The length of 𝐴�̂� = 40𝜋 cm

What is the length of 𝑆�̂�? What is the diameter? and 𝐶𝐴 ⃡ ∥ 𝐸𝑅 ⃡ . What is the

radius?

9) Find the angle formed between the minute hand and the hour hand when a clock is 12:30

10) Find the angle formed between the minute hand and the hour hand when a clock reads 10:20.

The traceries surrounding rose windows in Gothic cathedrals were constructed with

only arcs and straight lines. The photo (at right) shows a rose window from Reims

cathedral, which was built in the thirteenth century in Reims, a city in northeastern

France. The overlaid diagram shows its constructions.

11) Using the information above and the diagram reproduce the construction with

a compass and straight edge in the space below.

©Y f2z0a1d6m `KCuFtEah eSJoOfwttwfaXreez lLSLgCR.[ e mA_lilR crcihglhstlsx srWeVs]evryvaexdY.n x VMIasdfeW \wOi\tnhs pIUnafFilntiqtzeX WGTefoemJeltarRyA.

Worksheet by Kuta Software LLC

Geometry

10.9 Take Home Practice for Circle Exam

Name___________________________________ ID: 1

Date________________ Period____

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-1-

Find the length of each arc.

1)

6 cm

225°

2)

15 yd

150°

3)

12 m90°

Find the measure of the arc or central angle indicated. Assume that lines which appear to bediameters are actual diameters.

4) FEGm

F

G

HI

-25x + 5

-14x + 3

75°-52x + 4 E

5) mWX

UV

W

X

x + 166

x + 106

40°

60°

Find the perimeter of each polygon. Assume that lines which appear to be tangent are tangent.

6)

13

9.5

24.9

7) 13

23.1

20.7

8.3

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Worksheet by Kuta Software LLC

-2-

Find the measure of the arc or angle indicated.

8) Find mDE

C

D

E

11x + 11

7x + 7

9) Find mXY

W

X

Y6x + 16

15x + 5

10) Find mRST

R

ST

Vx + 103

x + 77

Use a compass and straight edge to perform the required constructions.

11) Construct the following polygons inscribed in a circle: 1) A square. 2) A regular Hexagon. 3) Anequilateral triangle.