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Circle Review
Geometry 6.1
Name each part of the circle below:
1. Name two chords.
2. Name one diameter.
3. Name three radii.
A
4. Name one tangent.
5. Name two semicircles.
6. Name nine minor arcs.
7. Name one major arc.
8. Draw a circle concentric to circle F.
D
C
B
G
E
F
H
Central Angles have vertices at the center of the circle and endpoints on
its circumference.
Name all of the central angles in circle C below:
B
D
C
E
A
Central angles and the arcs they define have equal measures.
Inscribed Angles have vertices and endpoints on a circle’s circumference.
Name all of the inscribed angles in circle C below:
F
D
C
B
A
E
Chord Properties
Geometry 6.1
Explain why: (informal proof)
Congruent chords define congruent arcs/central angles.
E
G
J
F
C
L
K
D
H
If two chords on the same circle are congruent, then the arcs and central
angles defined by them will be _____________.
Explain why: (informal proof)
The radius perpendicular to a given chord bisects it.
G
J
L
K
Is the converse true?
Explain why: (informal proof)
Congruent chords are equidistant from the center of the circle.
Given: Congruent
L
chords JL and PR.
K
J
M
R
N
P
Tangent Properties
Geometry 6.2
1. Construct a circle with center C.
Draw radius CX so that X is the midpoint of segment CD.
Construct the perpendicular bisector to segment CD through point X and
label it line AB.
D
A
X
C
B
AB should be tangent to circle C. Complete the satement below about
tangent lines:
A tangent to a circle will be ________ to the radius drawn to the point of
tangency.
Try the opposite:
2. Construct tangent line FG on circle C so that Y is the point of tangency.
Construct a line perpendicular to line FG through Y.
Does the line you constructed pass through the circle center?
F
Y
G
C
3. Construct circle K.
Construct tangent lines AL and AM with L and M on circle K.
What do you notice about Segments AL and AM?
L
K
A
M
Name________________________ Period _____
Chords and Tangents Practice
Geometry 6.2
Determine the length or measure of each x labeled below:
1.
2.
W
E
U
25o
Y
xo
V
40o
xo
D
C
F
215o
Z
3.
4.
E
W
o
x
35o
V 95
o
Y x
o
F
C
D
Z
Determine the length or measure of each x labeled below:
H
5.
6.
T
R xo
E
E
54o
xo
A
C
F
B
40o
8.
cm
60
m
xc
C
10
0c
m
70o
cm
9c
m
x
6c
m
7.
D
Arcs and Angles
Geometry 6.3
Inscribed Angles
Compare the measure of the inscribed angle below to the intercepted arc.
E
80o
xo
D
F
C
The same holds true for all angles inscribed in a circle (the proof is more
difficult when the angle does not pass through the circle center).
Find each angle measure x:
Z
1.
E
2.
110o
3.
E
xo
120o
xo
C
Y
xo
F
C
D
C
40o
190o
W
F
D
Based on figure three, we can conclude that:
Inscribed angles that intercept the same arc are _________.
Consider the following special cases.
Find each angle measure x:
Z
140o
E
x Y
xo
150o
N
xo 30o
F
o
C
C
D
X
Angles inscribed in a semicircle are always ______.
M
C
L
Arcs and Angles
Geometry 6.3
Look at the inscribed quadrilateral below.
What relationship can you discover about the angles?
E
1.
E
2.
110o
40 o
D
70o
140o
F
C
F
C
40o
30o
220o
G
G
D
70o
These quadrilaterals are called cyclic quadrilaterals.
In cyclic quadrilaterals, ________ angles are ________.
Finally, what can you conclude about the intercepted arcs in the
figure below?
E
xo
D
F
C
yo
G
Find each angle measure x:
1.
2.
130o
310o
N
E
xo
Q
D
F
C
70o
C
S
90o
G
R
o
x
xo
M
L
Name________________________ Period _____
Arcs and Angles Practice
Geometry 6.3
Determine the length or measure of each x labeled below:
90o
9.
V
160o
10. F
W
xo
75
C
o
C
D
U
Y
E
xo
11.
12.
E
80o
D
C
D
xo E
G
A
50o
F
xo
H
C
B
140o
Determine the length or measure of each x labeled below:
13.
T
R
E
5cm
A
H
G
10cm
F
xc
m
7cm
E
105o
14.
H
C
xo
J
B
o
E 25 C
15.
xo
L
E
16.
D
78o L
xo
B
C
A
B
W
58o
A
Name________________________ Period _____
Practice Quiz: Circle Properties
Geometry 6.3
Solve for x. Drawings not necessarily to scale.
1.
2.
W
U
xo
C
90o
C
xo
B
Y
x=_____
3.
D
x=_____
200o
4.
E
90o
D
F
C
C
D
44o
xo
xo E
G
B
5.
130o
x=_____
6.
T
90o
x=_____
E
xo
E
xo
70o
A
R
F
H
C
D
B
7.
x=_____
8.
E
70
xo
o
D
C
B
R
x=_____
E
L
C
xo
A
x=_____
22o
A
x=_____
Name________________________ Period _____
Practice Quiz: Circle Properties
Geometry 6.3
Solve for x: Drawings not necessarily to scale.
130o
9.
C
U
D
10.160o
W
90o
E
xo
F
C
o
x
B
Y
V
11.
x=_____
x=_____
12.
D
E
F
o
x
26o
E
C
xo
D
85o
C
G
B
o
120
x=_____
x=_____
C
18cm
x cm
B
F
14.
C
G
23c
m
F
A
H
J
12cm
M
xc
m
8cm
L
16c
m
13.
K
W
x=_____
x=_____
Geometry 6.4
Circle Proofs
Some Algebraic Properties that are useful when applied to Geometric
Proofs:
Properties of equality:
Addition, subtraction, multiplication, and division properties of equality:
Substitution:
Transitive Property:
Reflexive Property:
We have been using the Inscribed Angle Conjecture:
The measure of an angle inscribed in a circle is half the measure of the
intercepted arc.
Prove it:
Use:
Central Angle Conjecture,
Exterior Angle Conjecture (for triangles),
Base angles of an isosceles triangle,
Substitution.
D
Show:
m DFE = 1/2 mDE
E
y
x
z
C
F
Use the Inscribed Angle Conjecture
to prove the two cases below:
F
a
b
E
C
y
x
G
E
b
D
c
F
z
y
C
x
G
a
D
Name/label everything (given)
Use Algebra.
Show m GDF = 1/2 mDF
Show m EGF = 1/2 mEF
Name________________________ Period _____
Proofs Practice
Geometry 6.4
Prove each of the following using a two-column proof:
~ BE
1. Prove DE =
A
a
D
C
c b
d
B
E
hint: This proof relies heavily
on the transitive property.
2. Prove AB bisects CD at X
C
B
D
X
E
A
hint: You may use the
Tangent Segments Conjecture (p. 314).
3. Prove that
1
x  ( z  y)
2
D
B
A
x
y
z
C
E
hint: Add BE to the diagram.
Name________________________ Period _____
More Cool Circle Properties
4. Solve:
AB is tangent to circle E at C.
Arc CD = 100o
B
Find m BCD
C
A
E
D
hint: Add diameter CF.
5. Solve:
AB is tangent to circle O at C and AD is tangent at E.
Arc CE = 100o
B
Find m A
C
A
100o
O
E
D
6. Solve:
AB is tangent to circle O at C.
Arc CD = 130o and arc CE=60o
B
Find m A
C
O
A
E
hint: Add chord CD. Refer to #4.
D
Geometry 6.4
Pi
Geometry 6.5
Pi is the ratio of a circle’s circumference to its diameter.
3.14159 26535 89793 23846 26433 83279 50288 41971 69399 37510
Pi is computed to millions of digits using a variety of techniques like the
following continued fraction:
Here is another method for computing Pi, computed by Indian Mathematician Srinivasa Ramanujan (1887-1920 look him up).
This is the formula that is the basis for most computer estimations of Pi to
millions of digits. Looks easy, right?
Complete the following exercises involving Pi before we move on to
some more difficult problems:
1.
2.
3.
4.
Find the radius of a circle whose circumference is 6
What is the circumference of a circle whose radius is 1/2?
Find the circumference of a circle whose radius is 4cm in terms of pi.
A semicircle has a diameter of 10 inches. What is the perimeter of the
entire figure?
10 in
5. The radius of an automobile tire is 22 inches. The tire has a 40,000
mile warranty. How many rotations will the tire make before the
warranty expires? Use the pi button on your calculator to estimate
to the nearest whole number of rotations. (1 mile = 5,280 feet)
Pi
Geometry 6.6
Find each perimeter in terms of Pi:
1. The perimeter of the square
formed by connecting the
radii is 64cm.
Find the perimeter around the
outside of the figure.
2. The circles are congruent
and have a 3 inch radius.
Find the perimeter around the
outside of the figure.
Solve the following word problems involving Pi and speed:
1. The radius of the Earth at the equator is 3,963 miles. The earth makes
one rotation every 24 hours. If you are standing on the equator, you are
spinning around the Earth at a very high speed. Standing on the poles, you
are only rotating. How fast are you ‘moving’ around the center of the Earth
while standing on the equator (to the nearest mph).
2. Kimberly is jumping rope. The rope travels in a perfect circle with a
diameter of 7 feet. She can skip 162 times in a minute if she really tries.
How fast is the center of the jump rope traveling as she jumps rope?
3. A compact disc has a radius of approximately 2.25 inches. Your highspeed disc burner spins the CD at 8,000 RPM (revolutions per minute).
What is the speed at the edge of the disc to the nearest mile per hour?
Challenge: Approximate the radius of orbit for the moon.
1. The moon takes approximately 28 days to travel around the Earth. It is
traveling at a relative speed of about 2,230 mph around the earth. What is
the radius of the moon’s orbit (Based on the statistics given, what is the
distance from the Earth to the moon to the nearest mile?).
Geometry 6.7
Arc Length
Determine the arc length for each figure below in terms of pi:
CD = _____
CD = _____
C
CD = _____
C
3cm
C
E 12cm
D
E
4cm
E
D
B
36o
B
D
Arc length can be found using a fraction of the circumference.
Determine the arc length for each figure below in terms of pi:
CD = _____
CD = _____
C
CD = _____
C
72o
D
C
80o
12cm
B
4cm
80
o
B
E
E
3cm
35o
D
D
A
Work in reverse:
Find each radius.
1.
2.
6
C
3.
C
D
55o
C
3
25
B
D
50o
E
E
30
o
D
B
A
A
Geometry 6.7
Review
What have we learned?
Conjectures: (Theorems)
Chords Arcs Conjecture: Congruent Chords define Congruent Arcs.
Chord Perpendicular Bisectors: The perpendicular bisector of a chord
passes through the center of the circle.
Tangent Conjecture: Tangents are Perpendicular to Radii.
Tangent Segments Conjecture: Intersecting tangent lines are
congruent.
Inscribed Angle Conjecture: Inscribed angles are half the measure of
intercepted arcs.
Cyclic Quadrilaterals Conjecture: Opposite angles are supplementary.
Parallel Lines Conjecture: Intercepted arcs of parallel lines are
congruent.
C
F
B
Can you prove CF BD?
~ BG?
Can you prove BD =
A
D
E
H
G
~ CD?
Can you prove BC =
How do you know
supplementary?
BGH and
How do you know ACF is a right angle?
Can you prove that
CAD is a right angle?
FCD = 32o. Find the measures of all minor arcs in the figure.
What is the measure of H?
BC = 16 inches. What is the radius of circle E?
HDB are
Name________________________ Period _____
Review
Geometry
Write a relationship demonstrated by each diagram.
Example:
a
1
a b
2
b
a
a
b
a
b
a
a
a
b
b
b
a
a
b
b
a
b
c
Review
Geometry 6.7
Find the missing angle measure x for each figure below.
1.
2.
F
C
150o
S
F
L
C
75o A
xo
A
xo
J
T
A
P
Find the missing arc measure x for each figure below.
1.
2.
F
125o
W
R
xo
A
U
C
xo
C
D
L
80o
A
46o
E
T
Find the missing length x for each figure below.
1.
Challenge:
6
Q
x
R
x
C
A
40cm
S
E
hint: a2+b2=c2
leave answer in radical form
60cm
hint: a2+b2=c2
Pythagorean Introduction
Geometry
The Pythagorean Theorem is useful in solving all sorts of problems. Because
right angles appear so often in circles, it is necessary to give a brief introduction
to the Pythagorean Theorem before moving farther.
Pythagorean Theorem:
The sum of the squares of the legs of a right triangle is equal to the square of
the hypotenuse, or more familiar to most: a 2  b 2  c 2
ten
use
Ex.
13 c
5cm
Leg (a)
Hy
po
( c)
m
12cm
Leg (b)
Practice: Solve for x in each.
Leave answers in radical form where applicable.
1.
15cm
2.
25
cm
x
3.
8cm
x
8cm
x
7cm
7cm
Remember how to simplify Square Roots:
1.
2.
12
3.
72
5
2
2 cm
Now, find the area of each circle in terms of Pi:
1.
2.
3.
ABC is equilateral.
with AC=8cm
B
4c m
m
15c
1 6c m
9 cm
A
D
C
Name________________________ Period _____
Pythagoras and the Circle
Geometry 6.7
Solve for x in each of the following diagrams:
1.
2.
5c
m
2cm
x
2cm
x
x ________
3.
x ________
9cm
4.
5cm
4cm
4cm
x
45o
x
x ________
x ________
Find the area of each circle below: Circle Area =
5.
 r2
B
6.
2cm
m
4c
A
AB=10cm
m
6c
 cm2
Circle Area ________
 cm2
Circle Area (each) ________
Name________________________ Period _____
Pythagoras and the Circle
Geometry 6.7
Find the shaded area in each diagram below:
Answers should be expressed in terms of Pi and in simplest radical form where
appllcable.
2cm
7.
4cm
Area ___________ cm2
8.
12cm
45o
Area ___________ cm2
9.
X
B
Y
The shaded areas are called ‘lunes’.
ABC, AXB, and BYC are all semicircles.
AB = 16, BC=30. Find the combined
area of the shaded lunes.
C
A
Area ___________ cm2
10.
Small circles of radius 5 are placed inside
a semicircle of radius 18 so that the smaller
circles are tangent to the large semicircle.
Find the shaded area.
A
B
C
For hint: Read this upside-down in a mirror.
Area ___________ cm2
Name________________________ Period _____
Circle Properties Practice Test
Geometry 6.7
Solve for x in each of the following diagrams:
1.
Z
2.
90
Y
C
W
C
A
V
50o
o
xo
Z
M
Y
L
xo
x ________
3.
R
x ________
R
4.
B
xo
o
50
xo
A
C
C
52o
G
H
B
N
A
x ________
x ________
Find the missing length for each figure below (in terms of pi where applicable):
5.
6.
x
C
11
45o
54o
C
55o
x
6cm
x ________
x ________
7. The tire of a car traveling 30 mph makes 500 rotations per minute.
What is the radius of the tire? note: 1 mile = 5,280 feet.
Round to the nearest inch.
7. _______
Name________________________ Period _____
Circle Properties Practice Test
Geometry 6.7
Find the perimeter of each figure:
8.
9.
15cm
6cm
perimeter of the figure: ________
10cm
6cm
perimeter of the figure: ________
10. Prove:
~ BED.
In the figure below, prove that ADB =
A
B
C
F
You may use the conjecture names or brief
descriptions to justify each step.
Remember to list given information.
E
D
____________________________
____________________________________
____________________________
____________________________________
____________________________
____________________________________
____________________________
____________________________________
____________________________
____________________________________
____________________________
____________________________________
____________________________
____________________________________
____________________________
____________________________________
____________________________
____________________________________
Name________________________ Period _____
Geometry 6.7
Circle Properties Practice Test
Find the perimeter of each figure:
8.
9.
15cm
6cm
perimeter of the figure: ________
10cm
6cm
perimeter of the figure: ________
Find the area of each shaded region (in terms of pi where applicable):
X
10.
11. AB = 2 5
B
A
W
B
20cm
Y
A
(this one is a little harder than the
real test question, but you should
be able to do it)
total shaded area: ________
total shaded area: _______
Name________________________ Period _____
Geometry 6.7
Review Problem Set
1.
Find the missing angle measure x:
R
xo
A
_____
C
G
50o
B
2.
Find the missing length x
in terms of pi:
x
H
30o
C
20cm
_____
T
G
50o
F
L
110o
3.
Find the missing radius:
_____
E
I
2 cm
T
120o
75o
C
S
x
R
190o
4.
Find the perimeter
if the congruent circles
have radii of 5cm:
_____
Name________________________ Period _____
Geometry 6.7
Challenge Problem Set
5.
Find the measure
of arc XB.
Y
A
_____
D
20o
80o C
X
B
6.
Find the measure
of angle BDE.
C
B
xo D
_____
A
40o
G
E
F
7.
Segment AB is tangent to circle C at point B. If AB = 8cm and the circle radius is 6cm,
what is the shortest possible distance AX where X is a point on the circle?
8.
What is the area of the circle inscribed within an isosceles trapezoid whose bases are
9cm and 16cm long? Express your answer in terms of pi.
Name________________________ Period _____
Geometry 6.7
Circle Properties
Practice
Solve for x in each:
1.
2.
100o
L
50o
D
xo F
W
L
M
C
110o
o
C
x
Y
A
N
x = ________
3.
45o
x ________
4.
R
B
D
xo
75o
E
H
F
C
E
36o
xo
78o
C
S
85o
A
G
H
x ________
x ________
Find the missing length for each figure below (in terms of pi where applicable):
5.
6.
A
9
O
E
6cm
C
46o
x
135o x
M
105o
I
T
110o
x ________
x ________
Name________________________ Period _____
Geometry 6.7
Circle Properties
Practice:
Solve each.
7. The track coach at UNC wants an indoor track that is banked and a perfect circle,
sized so that if a runner can finish one lap in one minute, he will be traveling at 15mph.
What should the radius of the track be? (to the nearest foot, 5280ft = 1mile.)
7. _______
Find the perimeter of each:
8.
The perimeter of the interior
square is 11 inches.
9.
1cm
perimeter of the figure: ________
(to the nearest tenth)
5cm
5cm
1cm
perimeter of the figure: ________
(in terms of pi)
Name________________________ Period _____
Geometry 6.7
Circle Properties Retake Work
Practice
Solve for x in each:
1.
2.
100o
D
L
50o
W
L
M
xo F
xo
C
50o
Y
A
N
80
o
x = ________
3.
x ________
4.
R
96o
B
D
xo
F
E
o
20
H
xo
C
C
36o
S
E
A
x ________
x ________
Find the missing length for each figure below (in terms of pi where applicable):
A
5.
70o
110
8
6.
O
o
C 3cm
x
E
M
I
80
x
o
T
x ________
60o
x ________
7. A racetrack has two semi-circular turns, each with a radius of 1000ft, and two straightaways, each 2000 feet long. To the nearest second, how long will a car traveling 120 miles
per hour take to complete one lap on the track?
7. _______
Name________________________ Period _____
Circle Properties Retake Work
Geometry 6.7
Find the perimeter of each figure (outside the figure only):
8.
9.
A
10
cm
B
C
B
4cm
A
C
26cm
(Hexagon ABCDEF is regular.)
(Triangle ABC is equilateral.)
perimeter of the figure: ________
(to the nearest tenth)
perimeter of the figure: ________
(in terms of pi)
Find the area of each shaded region (in terms of pi where applicable):
11.
2c
m
10.
C
18cm
The vertices of the smaller square
are the midpoints of the larger
square, which is inscribed in
circle C.
total shaded area: ________
total shaded area: _______