10.1 circles and circumference

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.

A circle with center P is called “circle P” or P.

P

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.

center

diameter

radius

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 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.

P

Q

R

S

Name the circle.

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

Example 1a:

Answer: Four radii are shown: .

Name the radius of the circle.

Example 1b:

Answer: Four chords are shown: .

Name a chord of the circle.

Example 1c:

Name a diameter of the circle.

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

Example 1d:

Answer:

Answer:

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:

Your Turn:

Answer: 9

Formula for radius

Substitute and simplify.

If ST 18, find RS.Circle R has diameters and .

Example 2a:

Answer: 48

Formula for diameter

Substitute and simplify.

If RM 24, find QM.Circle R has diameters .

Example 2b:

Answer: So, RP = 2.

Since all radii are congruent, RN = RP.

If RN 2, find RP.Circle R has diameters .

Example 2c:

Answer: 58

Answer: 12.5

a. If BG = 25, find MG.

b. If DM = 29, find DN.

Circle M has diameters

c. If MF = 8.5, find MG.

Answer: 8.5

Your Turn:

Find EZ.

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

Example 3a:

Since the diameter of FZ = 5.

Since the diameter of , EF = 22.

Segment Addition Postulate

Substitution

is part of .

Simplify.

Answer: 27 mm

Example 3a:

Find XF.

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

Example 3b:

Since the diameter of , EF = 22.

Answer: 11 mm

is part of . Since is a radius of

Example 3b:

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

b. Find EB.

Answer: 6.5 in.

Answer: 13.5 in.

Your Turn:

Circumference The circumference of a circle is

the distance around the circle. In a circle,

C = 2r or d

Find C if r = 13 inches.

Circumference formula

Substitution

Answer:

Example 4a:

Find C if d = 6 millimeters.

Circumference formula

Substitution

Answer:

Example 4b:

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

Circumference formula

Substitution

Use a calculator.

Divide each side by .

Example 4c:

Radius formula

Use a calculator.

Answer:

Example 4c:

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.

Answer:

Answer:

Answer:

Your Turn:

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.

MULTIPLE- CHOICE TEST ITEM Find the exact circumference of . A B C D

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

Divide each side by 2.

Simplify.

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

Example 5:

Answer: C

Find the exact circumference of . A B C D

Your Turn:

Assignment Geometry

Pg. 526 #16 – 42 all, 44 – 54 evens

Pre-AP Geometry Pg. 526 #16 - 56