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Columbus, Ohio 43240
Lesson 10-1
Circles and Circumferences
Lesson 10-2
Angles and Arcs
Lesson 10-3
Arcs and Chords
Lesson 10-4
Inscribed Angles
Lesson 10-5
Tangents
Lesson 10-6
Secants, Tangents, and Angle Measures
Lesson 10-7
Special Segments in a Circle
Lesson 10-8
Equations of Circles
Example 1 Identify Parts of a Circle
Example 2 Find Radius and Diameter
Example 3 Find Measures in Intersecting Circles
Example 4 Find Circumference, Diameter, and Radius
Example 5 Use Other Figures to Find Circumference
Name the circle.
Answer: The circle has its center at E, so it is named
circle E, or
.
Name the radius of the circle.
Answer: Four radii are shown:
.
Name a chord of the circle.
Answer: Four chords are shown:
.
Name a diameter of the circle.
Answer:
are the only chords that go
through the center. So,
are
diameters.
a. Name the circle.
Answer:
b. Name a radius of the circle.
Answer:
c. Name a chord of the circle.
Answer:
d. Name a diameter of the circle.
Answer:
Circle R has diameters
and
.
If ST 18, find RS.
Formula for radius
Substitute and simplify.
Answer: 9
Circle R has diameters
.
If RM 24, find QM.
Formula for diameter
Substitute and simplify.
Answer: 48
Circle R has diameters
.
If RN 2, find RP.
Since all radii are congruent, RN = RP.
Answer: So, RP = 2.
Circle M has diameters
a. If BG = 25, find MG.
Answer: 12.5
b. If DM = 29, find DN.
Answer: 58
c. If MF = 8.5, find MG.
Answer: 8.5
The diameters of
and
are 22
millimeters, 16 millimeters, and 10 millimeters,
respectively.
Find EZ.
Since the diameter of
, EF = 22.
Since the diameter of
FZ = 5.
is part of
.
Segment Addition Postulate
Substitution
Simplify.
Answer: 27 mm
The diameters of
and
are 22
millimeters, 16 millimeters, and 10 millimeters,
respectively.
Find XF.
Since the diameter of
is part of
. Since
Answer: 11 mm
, EF = 22.
is a radius of
The diameters of
, and
are 5 inches,
9 inches, and 18 inches respectively.
a. Find AC.
Answer: 6.5 in.
b. Find EB.
Answer: 13.5 in.
Find C if r = 13 inches.
Circumference formula
Substitution
Answer:
Find C if d = 6 millimeters.
Circumference formula
Substitution
Answer:
Find d and r to the nearest hundredth if C = 65.4 feet.
Circumference formula
Substitution
Divide each side by .
Use a calculator.
Radius formula
Use a calculator.
Answer:
a. Find C if r = 22 centimeters.
Answer:
b. Find C if d = 3 feet.
Answer:
c. Find d and r to the nearest hundredth if C = 16.8 meters.
Answer:
MULTIPLE- CHOICE TEST ITEM Find the exact
circumference of
.
A
B
C
D
Read the Test Item
You are given a figure that involves a right triangle and
a circle. You are asked to find the exact circumference of
the circle.
Solve the Test Item
The radius of the circle is the same length as either leg
of the triangle. The legs of the triangle have equal
length. Call the length x.
Pythagorean Theorem
Substitution
Simplify.
Divide each side by 2.
Take the square root
of each side.
So the radius of the circle is 3.
Circumference formula
Substitution
Because we want the exact circumference, the answer
is B.
Answer: B
Find the exact circumference of
A
Answer: C
B
C
.
D
Example 1 Measures of Central Angles
Example 2 Measures of Arcs
Example 3 Circle Graphs
Example 4 Arc Length
ALGEBRA Refer to
Find
.
.
The sum of the measures of
Substitution
Simplify.
Add 2 to each side.
Divide each side
by 26.
Use the value of x to find
Given
Substitution
Answer: 52
ALGEBRA Refer to
Find
.
.
form a linear pair.
Linear pairs are supplementary.
Substitution
Simplify.
Subtract 140 from each side.
Answer: 40
ALGEBRA Refer to
a. Find m
Answer: 65
b. Find m
Answer: 40
.
In
Find
bisects
.
and
is a minor arc, so
is a semicircle.
is a right angle.
Arc Addition Postulate
Substitution
Subtract 90 from each side.
Answer: 90
In
Find
bisects
.
and
since
bisects
.
is a semicircle.
Arc Addition Postulate
Subtract 46 from each side.
Answer: 67
In
Find
bisects
.
and
Vertical angles are congruent.
Substitution.
Substitution.
Subtract 46 from each side.
Substitution.
Subtract 44 from each side.
Answer: 316
In
and
bisects
a.
Answer: 54
b.
Answer: 72
c.
Answer: 234
are diameters,
Find each measure.
and
BICYCLES This graph shows the percent of each type
of bicycle sold in the United States in 2001.
Find the measurement of the central angle representing
each category. List them from least to greatest.
The sum of the percents is 100% and represents the
whole. Use the percents to determine what part of the
whole circle
each central angle contains.
Answer:
BICYCLES This graph shows the percent of each type
of bicycle sold in the United States in 2001.
Is the arc for the wedge named Youth congruent to the
arc for the combined wedges named Other and
Comfort?
The arc for the wedge named Youth represents 26%
or
of the circle. The combined wedges named
Other and Comfort represent
. Since
º, the arcs
are not congruent.
Answer: no
SPEED LIMITS This graph shows the percent of U.S.
states that have each speed limit on their interstate
highways.
a. Find the measurement of the central angles
representing each category. List them from least to
greatest.
Answer:
b. Is the arc for the wedge for 65 mph congruent to the
combined arcs for the wedges for 55 mph and 70 mph?
Answer: no
In
and
. Find the length of
In
and
Write a proportion to compare each part to its whole.
.
.
degree measure of arc
degree measure of
whole circle
arc length
circumference
Now solve the proportion for .
Multiply each side by 9 .
Simplify.
Answer: The length of
is
units or about 3.14 units.
In
and
. Find the length of
Answer:
units or about 49.48 units
.
Example 1 Prove Theorems
Example 2 Inscribed Polygons
Example 3 Radius Perpendicular to a Chord
Example 4 Chords Equidistant from Center
PROOF Write a proof.
Given:
is a semicircle.
Prove:
Proof:
Statements
Reasons
1.
1. Given
is a semicircle.
2.
2. Def. of semicircle
3.
3. In a circle, 2 chords
are , corr. minor
arcs are .
4.
4. Def. of
5.
5. Def. of arc measure
arcs
Answer:
Statements
Reasons
6.
6. Arc Addition Postulate
7.
7. Substitution
8.
8. Subtraction Property
and simplify
9.
9. Division Property
10.
10. Def. of arc measure
11.
11. Substitution
PROOF Write a proof.
Given:
Prove:
Proof:
Statements
Reasons
1.
1. Given
2.
2. In a circle, 2 minor
arcs are , chords
are .
3.
3. Transitive Property
4.
4. In a circle, 2 chords
are , minor arcs
are .
TESSELLATIONS The rotations of a tessellation can
create twelve congruent central angles. Determine
whether
.
Because all of the twelve central angles are congruent,
the measure of each angle is
Let the center of the circle be A. The measure of
Then
.
The measure of
Then
.
Answer: Since the measures of
equal,
.
are
ADVERTISING A logo for an advertising campaign is a
pentagon that has five congruent central angles.
Determine whether
.
Answer: no
Circle W has a radius of 10 centimeters. Radius
is
perpendicular to chord
which is 16 centimeters
long.
If
find
Since radius
is perpendicular to chord
Arc addition postulate
Substitution
Substitution
Subtract 53 from each side.
Answer: 127
Circle W has a radius of 10 centimeters. Radius
is
perpendicular to chord
which is 16 centimeters
long.
Find JL.
Draw radius

A radius perpendicular to a
chord bisects it.
Definition of segment
bisector
Use the Pythagorean Theorem to find WJ.
Pythagorean Theorem
Simplify.
Subtract 64 from each side.
Take the square root of
each side.
Segment addition
Subtract 6 from each side.
Answer: 4
Circle O has a radius of 25 units. Radius
is
perpendicular to chord
which is 40 units long.
a. If
Answer: 145
b. Find CH.
Answer: 10
Chords
and
If the radius of
are equidistant from the center.
is 15 and EF = 24, find PR and RH.
are equidistant from P, so
.
Draw
to form a right triangle. Use the Pythagorean
Theorem.
Pythagorean Theorem
Simplify.
Subtract 144 from each side.
Take the square root of
each side.
Answer:
Chords
and
are equidistant from the center of
If TX is 39 and XY is 15, find WZ and UV.
Answer:
Example 1 Measures of Inscribed Angles
Example 2 Proofs with Inscribed Angles
Example 3 Inscribed Arcs and Probability
Example 4 Angles of an Inscribed Triangle
Example 5 Angles of an Inscribed Quadrilateral
In
and
Find the measures of the numbered angles.
First determine
Arc Addition
Theorem
Simplify.
Subtract 168 from
each side.
Divide each side
by 2.
So,
m
Answer:
In
and
measures of the numbered angles.
Answer:
Find the
Given:
Prove:
Proof:
Statements
Reasons
1.
1. Given
2.
2. If 2 chords are , corr.
minor arcs are .
3.
3. Definition of
intercepted arc
4.
4. Inscribed angles of
arcs are .
5.
5. Right angles are
congruent
6.
6. AAS
Given:
Prove:
Proof:
Statements
Reasons
1.
1. Given
2.
2. Inscribed angles of
arcs are .
3.
3. Vertical angles are
congruent.
4.
4. Radii of a circle are
congruent.
5.
5. ASA
PROBABILITY Points M and N are on a circle so
that
. Suppose point L is randomly located
on the same circle so that it does not coincide with
M or N. What is the probability that
Since the angle measure is twice the arc measure,
inscribed
must intercept
, so L must lie
on minor arc MN. Draw a figure and label any
information you know.
The probability that
is the same as the
probability of L being contained in
.
Answer: The probability that L is located on
is
PROBABILITY Points A and X are on a circle so
that
Suppose point B is randomly
located on the same circle so that it does not
coincide with A or X. What is the probability that
Answer:
ALGEBRA Triangles TVU and TSU are inscribed in
with
Find the measure of each
numbered angle if
and
are right triangles.
since
they intercept congruent arcs. Then the third angles of
the triangles are also congruent, so
.
Angle Sum Theorem
Simplify.
Subtract 105 from each side.
Divide each side by 3.
Use the value of x to find the measures of
Given
Answer:
Given
ALGEBRA Triangles MNO and MPO are inscribed
in
with
Find the measure of each
numbered angle if
and
Answer:
Quadrilateral QRST is inscribed in
find
and
Draw a sketch of this situation.
If
and
To find
To find
we need to know
first find
Inscribed Angle Theorem
Sum of angles in circle = 360
Subtract 174 from each side.
Inscribed Angle Theorem
Substitution
Divide each side by 2.
To find
find
we need to know
but first we must
Inscribed Angle Theorem
Sum of angles in circle = 360
Subtract 204 from each side.
Inscribed Angle Theorem
Divide each side by 2.
Answer:
Quadrilateral BCDE is inscribed in
find
and
Answer:
If
and
Example 1 Find Lengths
Example 2 Identify Tangents
Example 3 Solve a Problem Involving Tangents
Example 4 Triangles Circumscribed About a Circle
ALGEBRA
is tangent to
at point R. Find y.
Because the radius is perpendicular to the tangent at the
point of tangency,
. This makes
a right
angle and 
a right triangle. Use the Pythagorean
Theorem to find QR, which is one-half the length y.
Pythagorean Theorem
Simplify.
Subtract 256 from each side.
Take the square root of each
side.
Because y is the length of the diameter, ignore the
negative result.
Answer: Thus, y is twice
.
is a tangent to
Answer: 15
at point D. Find a.
Determine whether
is tangent to
First determine whether ABC is a right triangle by using
the converse of the Pythagorean Theorem.
Pythagorean Theorem
Simplify.
Because the converse of the Pythagorean Theorem did
not prove true in this case, ABC is not a right triangle.
Answer: So,
is not tangent to
.
Determine whether
is tangent to
First determine whether EWD is a right triangle by using
the converse of the Pythagorean Theorem.
Pythagorean Theorem
Simplify.
Because the converse of the Pythagorean Theorem is
true, EWD is a right triangle and EWD is a right angle.
Answer: Thus,
making
a tangent to
a. Determine whether
Answer: yes
is tangent to
b. Determine whether
Answer: no
is tangent to
ALGEBRA Find x. Assume that
segments that appear tangent to
circles are tangent.
are drawn from the
same exterior point and are
tangent to
so
are drawn from the same
exterior point and are tangent to
Definition of congruent segments
Substitution.
Use the value of y to find x.
Definition of congruent segments
Substitution
Simplify.
Subtract 14 from each side.
Answer: 1
ALGEBRA Find a. Assume that segments that
appear tangent to circles are tangent.
Answer: –6
Triangle HJK is circumscribed about
perimeter of HJK if
Find the
Use Theorem 10.10 to determine the equal measures.
We are given that
Definition of perimeter
Substitution
Answer: The perimeter of HJK is 158 units.
Triangle NOT is circumscribed about
perimeter of NOT if
Answer: 172 units
Find the
Example 1 Secant-Secant Angle
Example 2 Secant-Tangent Angle
Example 3 Secant-Secant Angle
Example 4 Tangent-Tangent Angle
Example 5 Secant-Tangent Angle
Find
Method 1
if
and
Method 2
Answer: 98
Find
if
Answer: 138
and
Find
Answer: 55
if
and
Find
Answer: 58
if
and
Find x.
Theorem 10.14
Multiply each side by 2.
Add x to each side.
Subtract 124 from each side.
Answer: 17
Find x.
Answer: 111
JEWELRY A jeweler wants to craft a pendant with
the shape shown. Use the figure to determine the
measure of the arc at the bottom of the pendant.
Let x represent the measure of
the arc at the bottom of the
pendant. Then the arc at the top
of the circle will be 360 – x. The
measure of the angle marked
40° is equal to one-half the
difference of the measure of the
two intercepted arcs.
Multiply each side by 2 and
simplify.
Add 360 to each side.
Divide each side by 2.
Answer: 220
PARKS Two sides of a fence to be built around a
circular garden in a park are shown. Use the figure
to determine the measure of
Answer: 75
Find x.
Multiply each side by 2.
Add 40 to each side.
Divide each side by 6.
Answer: 25
Find x.
Answer: 9
Example 1 Intersection of Two Chords
Example 2 Solve Problems
Example 3 Intersection of Two Secants
Example 4 Intersection of a Secant and a Tangent
Find x.
Theorem 10.15
Multiply.
Divide each side by 8.
Answer: 13.5
Find x.
Answer: 12.5
BIOLOGY Biologists often examine organisms under
microscopes. The circle represents the field of view
under the microscope with a diameter of 2 mm.
Determine the length of the organism if it is located
0.25 mm from the bottom of the field of view. Round to
the nearest hundredth.
Draw a model using a circle. Let x represent the unknown
measure of the equal lengths of the chord
which is
the length of the organism. Use the products of the
lengths of the intersecting chords to find the length of the
organism. Note that
Segment products
Substitution
Simplify.
Take the square root of each
side.
Answer: 0.66 mm
ARCHITECTURE Phil is installing a new window in an
addition for a client’s home. The window is a rectangle
with an arched top called an eyebrow. The diagram
below shows the dimensions of the window. What is
the radius of the circle containing the arc if the
eyebrow portion of the window is not a semicircle?
Answer: 10 ft
Find x if EF 10, EH 8, and FG 24.
Secant Segment Products
Substitution
Distributive Property
Subtract 64 from each side.
Divide each side by 8.
Answer: 34.5
Find x if
Answer: 26
and
Find x. Assume that segments that appear to be
tangent are tangent.
Disregard the negative solution.
Answer: 8
Find x. Assume that segments that appear to be
tangent are tangent.
Answer: 30
Example 1 Equation of a Circle
Example 2 Use Characteristics of Circles
Example 3 Graph a Circle
Example 4 A Circle Through Three Points
Write an equation for a circle with the center at
(3, –3), d 12.
Equation of a circle
Simplify.
Answer:
Write an equation for a circle with the center at
(–12, –1), r 8.
Equation of a circle
Simplify.
Answer:
Write an equation for each circle.
a. center at (0, –5), d 18
Answer:
b. center at (7, 0), r 20
Answer:
A circle with a diameter of 10 has its center in the
first quadrant. The lines y –3 and x –1 are
tangent to the circle. Write an equation of the circle.
Sketch a drawing of the two
tangent lines.
Since d 10, r 5. The line x –1 is perpendicular to a
radius. Since x –1 is a vertical line, the radius lies on a
horizontal line. Count 5 units to the right from x –1.
Find the value of h.
Likewise, the radius perpendicular to the line y –3 lies
on a vertical line. The value of k is 5 units up from –3.
The center is at (4, 2),
and the radius is 5.
Answer: An equation for the circle is
.
A circle with a diameter of 8 has its center in the
second quadrant. The lines y –1 and x 1 are
tangent to the circle. Write an equation of the circle.
Answer:
Graph
Compare each expression in the equation to the standard
form.
The center is at (2, –3), and the radius is 2.
Graph the center. Use a compass set at a width
of 2 grid squares to draw the circle.
Answer:
Graph
Write the expression in standard form.
The center is at (3, 0), and the radius is 4.
Draw a circle with radius 4, centered at (3, 0).
Answer:
a. Graph
Answer:
b. Graph
Answer:
ELECTRICITY Strategically located substations are
extremely important in the transmission and
distribution of a power company’s electric supply.
Suppose three substations are modeled by the
points D(3, 6), E(–1, 0), and F(3, –4). Determine the
location of a town equidistant from all three
substations, and write an equation for the circle.
Explore You are given three points that lie on a circle.
Plan
Graph DEF. Construct the perpendicular
bisectors of two sides to locate the center, which
is the location of the tower. Find the length of a
radius. Use the center and radius to write an
equation.
Solve
Graph DEF and construct the perpendicular
bisectors of two sides. The center appears to be
at (4, 1). This is the location of the tower. Find r
by using the Distance Formula with the center
and any of the three points.
Write an equation.
Examine You can verify the location of the center by
finding the equations of the two bisectors and
solving a system of equations. You can verify
the radius by finding the distance between the
center and another of the three points on the
circle.
Answer:
AMUSEMENT PARKS The designer of an
amusement park wants to place a food court
equidistant from the roller coaster located at (4, 1),
the Ferris wheel located at (0, 1), and the boat ride
located at (4, –3). Determine the location for the
food court and write an equation for the circle.
Answer:
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