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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 π΄πΆ and π΅πΆ. Then measure the angle β π΄πΆπ΅.
Step 4: What is the measurement of β π΄πΆπ΅?
Step 5: Repeat steps 2 β 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.
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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 ππΆ.
2) What kind of triangle is ΞAOC? _____________________ label the congruent angles βxβ
What kind of triangle is ΞOCB? _____________________ label the congruent angles βyβ
3) Since the sum of the angles in a triangle is 180Β°, express β π΄ππΆ in terms of x and β πΆππ΅ in terms of y.
β π΄ππΆ = _____________________ β πΆππ΅ = _____________________
4) What kind of angle pair is β π΄ππΆ πππ β πΆππ΅ ? _____________________
Write an equation relating β π΄ππΆ πππ β πΆππ΅. _______________________________________
5) Use your equations in steps 3 and 4 to algebraically prove that β πΆ is a right angle.
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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 β 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.
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DO EACH CONSTRUCTION AT LEAST 3 TIMES β 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.
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Name: ___________________________ Period: ________ 10.3 Circle Properties
Solve for the variable. Show your calculations. Mark the diagrams with measurements.
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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 itβs center?
Trace the crater and locate its center. Using the scale shown, find its radius. SHOW YOUR
WORK.
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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.
ππ
= 13 and ππ = 12.
What is the perimeter of POST?
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6. A satellite in geostationary orbit remains above the same point on the Earthβs surface even as the Earth
turns. If such a satellite has a 30Β° 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. π΄π β‘ is tangent to P at A(8,15). Find the equation of π΄π β‘ .
8. ππ΄ and ππ΅ are tangent to circle O. Something is wrong with this picture. Carefully explain what the
problem is.
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Name: ___________________________ Period: ________ 10.5 Apple Pi
3. What are sources of error for this experiment?
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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.
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Name: __________________________ Period: _________ 10.5 b Degrees to Radians
1) Given the radius of a circle is 1 unit, and the formula for circumference is πΆ = 2ππ, the distance
around the outside of the circle will be 2π. Use the circumference formula to indicate each of the
following distances in terms of π
2) The angle that creates the circumference of the circle can be measured in two ways.
A) Use the radius drawn in the circle on the left, what is the measure in degrees of the angle that you
would rotate the radius, from the center of the circle, until the radius maps back on to itself? __________
B) Label each of the angles in the circles, in degrees, for their indicated arc length.
C) That same angle around an entire circle that you measured in degrees, can be measured in radians. For
the circle on the left, that angle is 2π radians. Therefore 3600 = 2π πππππππ . You can use this
conversion to change between degrees and radians.
D) Label each of the angles in the circles, in radians, for their indicated arc length. Leave in terms of π.
2) πΌπ 3600 = 2π πππ£πππ πππβ π πππ ππ π‘βπ πππ’ππ‘πππ ππ¦ 2, π€βππ‘ ππ π¦ππ’ πππ‘?
When no units of angle measure are specified, radian measure is implied.
A) Convert the following degrees into radians. Show your work, leave your answer in terms of π.
5100 _____ 1200 _____ 2150 _____
B) Convert the following radians into degrees. Show your work.
5π
3 ___
2π
9 ___
41π
36 ___
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3) The largest a Pac-Man can open its mouth is 600, what is the largest it can
open it when measured in radians?
4) Convert between degrees or radians, show your work and leave your answer in terms of π when in
radians.
1500 _____ π
18 _____ 1650 _____
5π
4 _____ 6900 _____
14π
3 _____
0α΅ _____ Ο _____ 2 Ο _____
5) Given an angle in radian measure, how can you determine if the degree measure is less than or greater
than 180α΅ before you do the conversion?
6) Given an angle in radian measure, how can you tell if the degree measure is greater than 360α΅ before
you do the conversion?
7) With your calculator in the appropriate mode, find each value. Round to three decimal places.
sin (1500) _____ cos ( π
18 ) _____ tan (1650) _____
tan (5π
4) _____ cos (6900) _____ sin (
14π
3) _____
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Name: _______________________________Period: ____________ 10.6 Arcs and Angles
Find the indicated measure. SHOW YOUR WORK.
Hint: β what do you know about this line?
8. 9.
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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= _____
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Name: ______________________________ Period _____________ 10.7 Circumference Problems
1. Alfonzoβs 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 satelliteβs orbit has radius 1.59 Γ 106 m, calculate the satelliteβs 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 Γ 1011 m and Earth completes one revolution every
365.25 days, what is your βsittingβ 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 for circumference πΆ = 2 β ππ and speed π =π
π‘.
Thou shalt include thine own calculations. Thou shouldest convert thy units properly.
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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 38β 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 17β
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 moonβs 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.
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Name: __________________________ Period: ______ 10.8 Arc Length
1) What is the length of πΆοΏ½ΜοΏ½? 2) What is the length of πΈοΏ½ΜοΏ½?
3) What is the length of π΅πΌοΏ½ΜοΏ½? 4) If the length of π΄οΏ½ΜοΏ½ is 6π m, 5) The radius is 18 feet.
What is the radius? What is the length
of π
οΏ½ΜοΏ½?
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6) The radius is 9 m 7) The length of ποΏ½ΜοΏ½ is 12π 8) The length of π΄οΏ½ΜοΏ½ = 40π cm
What is the length of ποΏ½ΜοΏ½? What is the diameter? and πΆπ΄ β‘ β₯ πΈπ
β‘ . 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.
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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
225Β°
2)
15 yd
150Β°
3)
12 m90Β°
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
75Β°-52x + 4 E
5) mWX
UV
W
X
x + 166
x + 106
40Β°
60Β°
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.