10.1 Circles and Circumference

10.1 Circles and Circumference10.1 Circles and Circumference

ObjectivesObjectives

Identify and use parts of circlesIdentify and use parts of circles

Solve problems using the Solve problems using the circumference of circlescircumference of circles

Parts of CirclesParts of Circles

Circle Circle – set of all – set of all points in a plane points in a plane that are that are equidistant equidistant from a given point from a given point called the called the centercenter of of the circle.the circle.

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

P

Parts of CirclesParts of Circles

The distance from the The distance from the center to a point on the center to a point on the circle is called the circle is called the radiusradius of the circle. of the circle.

The distance across the The distance across the circle through its center is circle through its center is the the diameter diameter of the of the circle. The diameter is circle. The diameter is twice the radius twice the radius d = 2rd = 2r or or r = ½ dr = ½ d).).

The terms The terms radius radius and and diameterdiameter describe describe segments as well as segments as well as measures.measures.

center

diameter

radius

Parts of CirclesParts of Circles

QPQP , , QSQS , and , and QRQR are radii. are radii. All radii for the same circle All radii for the same circle

are congruent.are congruent. PRPR is a is a diameterdiameter.. All diameters for the same All diameters for the same

circle are congruent.circle are congruent. A A chordchord is a segment is a segment

whose endpoints are whose endpoints are points on the circle. points on the circle. PSPS and and PRPR are chords. are chords.

A diameter is a chord that A diameter is a chord that passes through the center passes through the center of the circle.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:Example 1a:

Answer: Four radii are shown: .

Name the radius of the circle.

Example 1b:Example 1b:

Answer: Four chords are shown: .

Name a chord of the circle.

Example 1c:Example 1c:

Name a diameter of the circle.

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

Example 1d: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:Your Turn:

Answer: 9

Formula for radius

Substitute and simplify.

If ST 18, find RS.

Circle R has diameters and .

Example 2a:Example 2a:

Answer: 48

Formula for diameter

Substitute and simplify.

If RM 24, find QM.

Circle R has diameters .

Example 2b:Example 2b:

Answer: So, RP = 2.

Since all radii are congruent, RN = RP.

If RN 2, find RP.

Circle R has diameters .

Example 2c: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:Your Turn:

Find EZ.

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

Example 3a: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:Example 3a:

Find XF.

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

Example 3b:Example 3b:

Since the diameter of , EF = 22.

Answer: 11 mm

is part of . Since is a radius of

Example 3b: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:Your Turn:

CircumferenceCircumference

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

C = 2C = 2ππr or r or ππdd

Find C if r = 13 inches.

Circumference formula

Substitution

Answer:

Example 4a:Example 4a:

Find C if d = 6 millimeters.

Circumference formula

Substitution

Answer:

Example 4b: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:Example 4c:

Radius formula

Use a calculator.

Answer:

Example 4c: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: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: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: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:Example 5:

Answer: C

Find the exact circumference of .

A B C D

Your Turn:Your Turn:

AssignmentAssignment

GeometryGeometryPg. 526 #16 – 42, 44 – 54 Pg. 526 #16 – 42, 44 – 54

evensevens

Pre-AP Geometry Pre-AP Geometry Pg. 526 #16 - 56Pg. 526 #16 - 56