10.1 Circles and Circumference
Objectives
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Identify and use parts of circles
Solve problems using the circumference
of circles
Parts of Circles
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Circle – set of all
points in a plane
that are equidistant
from a given point
called the center of
the circle.
A circle with center
P is called “circle P”
or
P.
P
Parts of Circles
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The distance from the
center to a point on the
circle is called the
radius of the circle.
The distance across the
circle through its center
is the diameter of the
circle. The diameter is
twice the radius d = 2r
or r = ½ d).
The terms radius and
diameter describe
segments as well as
measures.
Parts of Circles
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QP , QS , and QR are radii.
All radii for the same circle are
congruent.
PR is a diameter.
All diameters for the same
circle are congruent.
A chord is a segment whose
endpoints are points on the
circle. PS and PR are chords.
A diameter is a chord that
passes through the center of
the circle.
Example 1a:
Name the circle.
Answer: The circle has its center at E, so it is named
circle E, or
.
Example 1b:
Name the radius of the circle.
Answer: Four radii are shown:
.
Example 1c:
Name a chord of the circle.
Answer: Four chords are shown:
.
Example 1d:
Name a diameter of the circle.
Answer:
are the only chords that go
through the center. So,
are
diameters.
Your Turn:
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:
Example 2a:
Circle R has diameters
and
.
If ST 18, find RS.
Formula for radius
Substitute and simplify.
Answer: 9
Example 2b:
Circle R has diameters
.
If RM 24, find QM.
Formula for diameter
Substitute and simplify.
Answer: 48
Example 2c:
Circle R has diameters
.
If RN 2, find RP.
Since all radii are congruent, RN = RP.
Answer: So, RP = 2.
Your Turn:
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
Example 3a:
The diameters of
and
are 22
millimeters, 16 millimeters, and 10 millimeters,
respectively.
Find EZ.
Example 3a:
Since the diameter of
, EF = 22.
Since the diameter of
FZ = 5.
is part of
.
Segment Addition Postulate
Substitution
Simplify.
Answer: 27 mm
Example 3b:
The diameters of
and
are 22
millimeters, 16 millimeters, and 10 millimeters,
respectively.
Find XF.
Example 3b:
Since the diameter of
is part of
. Since
Answer: 11 mm
, EF = 22.
is a radius of
Your Turn:
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.
Circumference
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The circumference of a circle is the
distance around the circle. In a circle,
C = 2πr
or
πd
Example 4a:
Find C if r = 13 inches.
Circumference formula
Substitution
Answer:
Example 4b:
Find C if d = 6 millimeters.
Circumference formula
Substitution
Answer:
Example 4c:
Find d and r to the nearest hundredth if C = 65.4 feet.
Circumference formula
Substitution
Divide each side by .
Use a calculator.
Example 4c:
Radius formula
Use a calculator.
Answer:
Your Turn:
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:
Example 5:
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.
Example 5:
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.
Example 5:
So the radius of the circle is 3.
Circumference formula
Substitution
Because we want the exact circumference, the answer
is B.
Answer: B
Your Turn:
Find the exact circumference of
A
Answer: C
B
C
.
D
Assignment
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Geometry
Pg. 526 #16 – 42, 44 – 54 evens
Pre-AP Geometry
Pg. 526 #16 - 56