10.1 Circles and Circumference
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10.1 Circles and Circumference. Objectives. Identify and use parts of circles Solve problems using the circumference of circles. Parts of Circles. Circle – set of all points in a plane that are equidistant from a given point called the center of the circle.

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10.1 Circles and Circumference

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10 1 circles and circumference

10.1 Circles and Circumference


Objectives

Objectives

  • Identify and use parts of circles

  • Solve problems using the circumference of circles


Parts of circles

Parts of Circles

  • 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 circles1

Parts of Circles

  • 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 circles2

Parts of Circles

  • 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 chordis 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

Answer: The circle has its center at E, so it is named circle E, or .

Example 1a:

Name the circle.


Example 1b

Answer: Four radii are shown: .

Example 1b:

Name the radius of the circle.


Example 1c

Answer: Four chords are shown: .

Example 1c:

Name a chord of the circle.


Example 1d

Answer: are the only chords that go through the center. So, are diameters.

Example 1d:

Name a diameter of the circle.


Your turn

a. Name the circle.b. Name a radius of the circle.

c. Name a chord of the circle.

d. Name a diameter of the circle.

Answer:

Answer:

Answer:

Answer:

Your Turn:


Example 2a

Circle R has diameters and .

If ST18, find RS.

Example 2a:

Formula for radius

Substitute and simplify.

Answer: 9


Example 2b

Circle R has diameters .

If RM24, find QM.

Example 2b:

Formula for diameter

Substitute and simplify.

Answer: 48


Example 2c

Circle R has diameters .

If RN2, find RP.

Example 2c:

Since all radii are congruent, RN=RP.

Answer: So, RP=2.


Your turn1

Circle M has diameters

a. If BG=25, find MG.

b. If DM=29, find DN.

c. If MF=8.5, find MG.

Your Turn:

Answer: 12.5

Answer: 58

Answer: 8.5


Example 3a

The diameters of and are 22 millimeters, 16 millimeters, and 10 millimeters, respectively.

Example 3a:

Find EZ.


Example 3a1

Since the diameter of , EF = 22.

Since the diameter of FZ = 5.

is part of .

Example 3a:

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:


Example 3b1

Since the diameter of , EF = 22.

is part of . Since is a radius of

Example 3b:

Answer: 11 mm


Your turn2

The diameters of , and are 5 inches, 9 inches, and 18 inches respectively.

a. Find AC.

b. Find EB.

Your Turn:

Answer: 6.5 in.

Answer: 13.5 in.


Circumference

Circumference

  • The circumference of a circle is the distance around the circle. In a circle,

    C = 2r or d


Example 4a

Answer:

Example 4a:

Find C if r=13 inches.

Circumference formula

Substitution


Example 4b

Answer:

Example 4b:

Find C if d=6 millimeters.

Circumference formula

Substitution


Example 4c

Divide each side by .

Example 4c:

Find dand r to the nearest hundredth if C = 65.4 feet.

Circumference formula

Substitution

Use a calculator.


Example 4c1

Answer:

Example 4c:

Radius formula

Use a calculator.


Your turn3

Answer:

Answer:

Answer:

Your Turn:

a. Find C if r = 22 centimeters.

b. Find C if d = 3 feet.

c. Find d and r to the nearest hundredth if C = 16.8 meters.


Example 5

MULTIPLE-CHOICE TEST ITEM Find the exact circumference of .

A B C D

Example 5:

Read the Test ItemYou are given a figure that involves a right triangle and a circle. You are asked to find the exact circumference of the circle.


Example 51

Example 5:

Solve the Test ItemThe 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 52

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 turn4

Find the exact circumference of .

A B C D

Your Turn:

Answer: C


Assignment

Assignment

  • GeometryPg. 526 #16 – 42 all,44 – 54 evens

  • Pre-AP Geometry Pg. 526 #16 - 56


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