GEOMETRY
CHAPTER 10
Circles
SPRING 2017
0
Learning Targets
•
•
Students will be able to identify and use parts of circles.
Students will solve problems involving the circumference of a circle.
Section 10.1 Notes: Circles and Circumference
Vocabulary,
Example Types
Definition, Pictures, and Examples
Circle
Radius
Chord
Diameter
Compare and
Contrast Chord,
Radius and
Diameter
1
Example 1:
a) Name the circle and identify a radius.
b) Identify a chord and a diameter of the circle.
Examples
labeling the
circle, radius,
chord and
diameter.
Example 2:
a) Name the circle and identify a radius.
You Try!
b) Which segment is not a chord?
Formula for
Radius
Formula for
Diameter
What is the
relationship
between radius
and diameter?
Example 3:
Examples given
the radius or
diameter, find the
length of a part of
the circle.
You Try!
a) If RT = 21 cm, what is the length of QV ?
Example 4:
If QS = 26 cm, what is the length of
RV ?
2
Circle Pair – Two
circles are
congruent…
Circle Pair – Two
circles are
similar…
Circle Pair –
Concentric
Circles
Two Circles can
intersect in two
different ways.
Example 5:
a) The diameter of
X
is 22 units, the diameter of
Y
is 16 units, and WZ = 5 units. Find XY.
Example given
two circles
intersecting:
b) The diameters of
 D ,  B , and  A
are 4 inches, 9 inches, and 18 inches. Find AC.
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Circumference
Pi(π)
Example 6:
Examples of
Circumference
a) CROP CIRCLES A series of crop circles was discovered in Alberta, Canada, on September 4, 1999. The
largest of the three circles had a radius of 30 feet. Find its circumference.
b) The Unisphere is a giant steel globe that sits in Flushing Meadows-Corona Park in Queens, New York. It has
a diameter of 120 feet. Find its circumference.
You
Try!
Example 7:
a) Find the diameter and the radius of a circle to the nearest hundredth if the circumference of the circle is 65.4
feet.
b) Find the radius of a circle to the nearest hundredth if its circumference is 16.8 meters.
Inscribed
Circumscribed
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c) Find the exact circumference of
K .
Summary!
On the circles,
draw the
characteristics.
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Learning Targets:
1.
2.
I can identify central angles and their measurements.
I can identify arcs and their labels.
Section 10.2 Notes: Measuring Angles and Arcs
Vocabulary,
Example Types
Definition, Pictures, and Examples
Central Angle
Sum of Central
Angles
Example 1:
a) Find the value of x.
Examples in
finding the
b) Find the value of x.
Arc
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Minor Arc
Major Arc
Semicircle
Example 2: Use the figure on the right.
a) WC is a radius of C . Identify
Then find its measure.

XZY as a major arc, minor arc, or semicircle.
Examples
identifying arcs.
 as a major arc, minor arc, or semicircle.
b) WC is a radius of C . Identify WZX
Then find its measure.
c) WC is a radius of C . Identify 
XW as a major arc, minor arc, or semicircle.
Then find its measure.
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Example 3:
On the given circle, identify a major arc, minor arc and a semicircle.
You
Try!
Theorem 10.1
Example 4:
.
a) BICYCLES Refer to the circle graph. Find mKL
.
b) BICYCLES Refer to the circle graph. Find mNJL
Adjacent Arcs
Arc Addition
Postulate
Example 5:
Examples using
Addition
Postulate
 in
a) Find mLHI
M .
 in
b) Find the mBCD
F .
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Arc length
Example 6:
Examples finding
arc length
You
Try!
a) Find the length of
 . Round to the nearest hundredth.
DA
b) Find the length of
 . Round to the nearest hundredth.
DA
1. Find the length of arc AB .
.
2. The diameter is 24 cm. Find the length of arc
B
CD
D
6 cm
120°
A
C
60°
Example 7:
a. A circle has an arc whose measure is 80° and whose length is
b. A circle has a circumference whose length is
25π .
88π .
What is the diameter of the circle?
Find the length of an arc whose central angle is
90° .
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c. Find the measure of the central angle of an arc if its length is
14π
and the radius is 18.
Summary!
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Learning Targets
•
•
Students will be able to recognize and use relationships between arcs and chords.
Students will be able to recognize and use relationships between arcs, chords, and diameters
Section 10.3 Notes: Arcs and Chords
Vocabulary,
Example Types
Definition, Pictures, and Examples
Chord
Example 1:
a) JEWELRY A circular piece of jade is hung from a chain by two wires around the stone.
Examples with
Chords
 = 90°. Find mJM
.
JM ≅ KL and mKL
b)
W
has congruent chords
Example 2:
 = 85°, find mTV
.
RS and TV . If mRS
a) ALGEBRA In the figure,
 ≅ YZ
 . Find WX.
 A ≅  B and WX
b) ALGEBRA In the figure,
 ≅ LM
 . Find LM.
G ≅  H and RT
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Theorem 10.3
Theorem 10.4
Example 3:
a) In

.
G , mDEF
= 150° . Find mDE
b) In
= 60° . Find mUX
.
 Z , mWUX
Example 4:
a) CERAMIC TILE In the ceramic stepping stone below, diameter
inches long. Find CD.
b) In the circle below, diameter
AB is 18 inches long and chord EF is 8
QS is 14 inches long and chord RT is 10 inches long. Find VU.
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Theorem 10.5
Example 5:
Examples using
theorem 10.5
Summary!
a) In  P , EF = GH = 24. Find PQ.
b) In  R , MN = PO = 29. Find RS.
In the circle provided, draw a chord which
is not a diameter using a bolded line and label it, a diameter using
a thin line and label it, and a radius using a dashed line and label it.
Identify 3 arcs that were created on the circle.
1. Find the value of x.
2. Find the value of x.
13
Learning Targets
• Students will be able to find areas of circles.
• Students will find areas of sectors of circles.
Section 11.3 Notes: Areas of Circles and Sectors
Vocabulary,
Example Types
Definition, Pictures, and Examples
In Lesson 10-1, you learned that the formula for the circumference C of a circle with radius r is given by
𝐶 = 2𝜋𝑟. You can use this formula to develop the formula for the area of a circle.
Below, a circle with radius r and circumference C has been divided into congruent pieces and then rearranged to
form a figure that resembles a parallelogram.
Area of a Circle
Examples finding
the area.
Example 1: An outdoor accessories company manufactures circular
covers for outdoor umbrellas. If the cover is 8 inches longer than the umbrella on
each side, find the area of the cover in square inches.
Example 2: Find the radius of a circle with an area of 58 square inches.
Area of a Sector
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Examples of Area
of a Sector
Example 3:
a) A pie has a diameter of 9 inches and is cut into 10 congruent slices. What is the area of one slice to the
nearest hundredth?
b) A pizza has a diameter of 14 inches and is cut into 8 congruent slices. What is the area of one slice to the
nearest hundredth?
You Try!
Area of a Sector Practice: Find the area of the shaded sector. Round to the nearest tenth
a)
1.
a.
b)
Calculate the sector area:
b.
c.
Extra Practice /
Summary!
2. The area of a circle is 225π square inches. Find the area of the sector whose central angle is
45 .
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3. The central angle of a sector is
60 and the area of the circle is 144π . What is the area of the sector?
4. A circle has a radius of 12. Find the area of the sector whose central angle is
120 .
5. Find the radius of a circle which has a sector area of 9π and whose central angle is
6. The central angle of a sector is
90 .
72 and the sector has an area of 5π . Find the radius.
7. Find the measure of the central angle of a sector if its area is 5π and the radius is 6.
8. If the area of a circle is 254 m2, find the circumference. Round to the nearest tenth.
9. Find
m
AB if the area of the sector is 11.2 ft2. Round to the nearest degree.
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10. Find the radius of the circle if the area of the shaded region is 78.3 ft2.
Round to the nearest tenth.
11. Find the exact area of the shaded region.
12. Find the area of the shaded region. Round to the nearest hundredth.
Helping You Remember: A good way to remember something is to explain it to someone else. Suppose Jimmy
is having trouble remembering which formula is for circumference and which is
for area. How can you help him out?
Written
Summary!
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Learning Targets
•
•
Students will be able to find measures of inscribed angles.
Students will be able to find measures of angles of inscribed polygons.
Section 10.4 Notes: Inscribed Angles
Vocabulary,
Example Types
Definition, Pictures, and Examples
Inscribed Angle
Intercepted Arc
Inscribed Angle
Theorem
Example 1:
Examples of
intercepted Arc
a) Find
m∠X .
b) Find

mYX
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c) Find
m∠C.
d) Find

mBC
Theorem 10.7
Example 2:
Examples of
Theorem 10.7
a) Find
m∠R.
b) Find
m∠I .
Theorem 10.8
Example 3:
a) Find
m∠B.
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b) Find
m∠D.
Example 4:
a) INSIGNIAS An insignia is an emblem that signifies rank, achievement, membership, and so on. The insignia
shown is a quadrilateral inscribed in a circle. Find
m∠S
and
m∠T .
b) INSIGNIAS An insignia is an emblem that signifies rank, achievement, membership, and so on. The insignia
shown is a quadrilateral inscribed in a circle. Find m∠N .
Summary!
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Writing in math A 45-45-90 right triangle is inscribed in a circle. If the radius of the circle is given, explain how
to find the lengths of the right triangle’s legs.
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Learning Targets
•
•
Students will be able to use properties of tangents.
Students will be able to solve problems involving circumscribed polygons.
Section 10.5 Notes: Tangents
Vocabulary,
Example Types
Definition, Pictures, and Examples
Tangent
Point of Tangency
Common Tangent
Example 1:
Example
identifying
common tangent
a) Using the figure to the right, draw the common tangents. If no common tangent exists, state no common
tangent.
b) Using the figure to the right, draw the common tangents. If no common tangent exists, state no common
tangent.
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Theorem 10.10
Example 2:
a)
KL is a radius of  K . Determine whether LM is tangent to  K . Justify your answer.
b)
XY is a radius of  X . Determine whether YZ is tangent to  X . Justify your answer.
Example 3:
a) In the figure,
WE is tangent to  D at W. Find the value of x.
b) In the figure,
IK is tangent to  J at K. Find the value of x.
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You
Try!
Determine if line AB is tangent to the circle.
a.
Theorem 10.11
Example 4:
a)
AC and BC are tangent to  Z . Find the value of x.
b)
MN and MP are tangent to Q . Find the value of x.
Circumscribed
Polygons
24
Example 5:
a) The round cookies are marketed in a triangular package to pique the consumer’s interest. If
∆QRS is
circumscribed about T , find the perimeter of ∆QRS .
b) A bouncy ball is marketed in a triangular package to pique the consumer’s interest. If ∆ABC is
circumscribed about G , find the perimeter of ∆ABC .
Summary!
Draw a circumscribed triangle
1. Determine whether each segment is tangent
Draw a inscribed triangle
2. Solve for x given PW and WQ are tangents.
to the given circle. Justify your answer.
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Learning Targets
•
•
Students will be able to write the equation of a circle.
Students will be able to graph a circle on the coordinate plane.
Section 10.8 Notes: Equations of Circles
Vocabulary,
Example Types
Definition, Pictures, and Examples
Equation of a
Circle in
Standard Form
Example 1:Graph the following circle:
a. (x - 3)2 + (y + 1)2 = 4
Examples given
an equation of a
circle
b. (x – 2)2 + (y – 5)2 = 9
c. (y + 4)2 + (x + 2)2 = 16
1
Given the
equation of a
circle, identify the
following.
Example 2: Find the center and radius of each equation.
b. x2 + (y – 3)2 = 18
a. (x + 3)2 + (y – 1)2 = 4
c. x2 – 4x + y2 + 6y = –9
d. 𝑥 2 + 𝑦 2 + 6x – 6y + 9 = 0
Example 3:
a) Write the equation of the circle with a center at (3, –3) and a radius of 6.
Examples of
Equation of a
circle
b) Write the equation of the circle graphed below.
c) Write the equation of the circle graphed below.
2
Example 4: Write the equation of the circle.
You
Try!
a.
b.
Example 5:
a) Write the equation of the circle that has its center at (–3, –2) and passes through (1, –2).
Examples given
the center and
point on the
circle.
b) Write the equation of the circle that has its center at (–1, 0) and passes through (3, 0).
3
Example 6:
a) 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, 1), and F(3, –4).
Determine the location of a town equidistant from all three substations, and write an equation for the circle.
b) 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.
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c) Jimmy Johns offers delivery within 5 miles of the restaurant. Marcus’ house is located 4 miles east and 5
miles south of Jimmy Johns. Carly’s house is located 1 mile west and 2 miles south of Jimmy Johns. Conner’s
house is located 3 miles east and 5 miles north of Jimmy Johns. If Jimmy Johns is located at (–2, 1), which
house(s) can get delivery?
d) Cell phone towers are periodically placed around App Town so that they give a signal that reaches within 4
miles of the tower’s location. One particular tower is located at (3, 4). Houses A, B, and C are located at (–1, 4),
(5, 3), and (2, 0). Which house(s) receive a signal from this tower?
5
Summary!
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