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11-2. Arcs and Chords. Up. Lesson Presentation. Lesson Quiz. Holt Geometry. Warm Up 1. What percent of 60 is 18? 2. What number is 44% of 6? 3. Find m WVX. 30. 2.64. 104.4. Objectives. Apply properties of arcs. Apply properties of chords. Vocabulary.

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11 2

11-2

Arcs and Chords

Up

Lesson Presentation

Lesson Quiz

Holt Geometry


11 2

Warm Up

1.What percent of 60 is 18?

2. What number is 44% of 6?

3. Find mWVX.

30

2.64

104.4


11 2

Objectives

Apply properties of arcs.

Apply properties of chords.


11 2

Vocabulary

central anglesemicircle

arcadjacent arcs

minor arccongruent arcs

major arc


11 2

A central angleis an angle whose vertex is the center of a circle. An arcis an unbroken part of a circle consisting of two points called the endpoints and all the points on the circle between them.


11 2

Writing Math

Minor arcs may be named by two points. Major arcs and semicircles must be named by three points.


11 2

mKLF = 360° – mKJF

mKLF = 360° – 126°

Example 1: Data Application

The circle graph shows the types of grass planted in the yards of one neighborhood. Find mKLF.

mKJF = 0.35(360)

= 126

= 234


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b. mAHB

Check It Out! Example 1

Use the graph to find each of the following.

a. mFMC

mFMC = 0.30(360)

= 108

Central is 30% of the .

= 360° – mAMB

= 0.10(360)

c. mEMD

mAHB

=360° –0.25(360)

= 36

= 270

Central is 10% of the .


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Adjacent arcs are arcs of the same circle that intersect at exactly one point. RS and ST are adjacent arcs.


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Find mBD.

mBC = 97.4

mCD = 30.6

mBD = mBC + mCD

Example 2: Using the Arc Addition Postulate

Vert. s Thm.

mCFD = 180 – (97.4 + 52)

= 30.6

∆Sum Thm.

mCFD = 30.6

Arc Add. Post.

= 97.4 + 30.6

Substitute.

Simplify.

= 128


11 2

mJKL

mKL = 115°

mJKL = mJK + mKL

Check It Out! Example 2a

Find each measure.

mKPL = 180° – (40 + 25)°

Arc Add. Post.

= 25° + 115°

Substitute.

= 140°

Simplify.


11 2

mLJN

mLJN = 360° – (40 + 25)°

Check It Out! Example 2b

Find each measure.

= 295°


11 2

Within a circle or congruent circles, congruent arcs are two arcs that have the same measure. In the figure STUV.


11 2

TV WS. Find mWS.

TV  WS

mTV = mWS

mWS = 7(11) + 11

Example 3A: Applying Congruent Angles, Arcs, and Chords

 chords have arcs.

Def. of arcs

9n – 11 = 7n + 11

Substitute the given measures.

2n = 22

Subtract 7n and add 11 to both sides.

n = 11

Divide both sides by 2.

Substitute 11 for n.

= 88°

Simplify.


11 2

GD  NM

GD NM

Example 3B: Applying Congruent Angles, Arcs, and Chords

C  J, andmGCD  mNJM. Find NM.

GCD NJM

 arcs have chords.

GD = NM

Def. of chords


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Example 3B Continued

C  J, andmGCD  mNJM. Find NM.

14t – 26 = 5t + 1

Substitute the given measures.

9t = 27

Subtract 5t and add 26 to both sides.

t = 3

Divide both sides by 9.

NM = 5(3) + 1

Substitute 3 for t.

= 16

Simplify.


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PT bisects RPS. Find RT.

mRT  mTS

Check It Out! Example 3a

RPT  SPT

RT = TS

6x = 20 – 4x

10x = 20

Add 4x to both sides.

x = 2

Divide both sides by 10.

RT = 6(2)

Substitute 2 for x.

RT = 12

Simplify.


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mCD = mEF

mCD = 100

Check It Out! Example 3b

Find each measure.

A B, and CD EF. Find mCD.

 chords have arcs.

Substitute.

25y = (30y – 20)

Subtract 25y from both sides. Add 20 to both sides.

20 = 5y

4 = y

Divide both sides by 5.

CD = 25(4)

Substitute 4 for y.

Simplify.


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Step 1 Draw radius RN.

RM NP , so RM bisects NP.

Example 4: Using Radii and Chords

Find NP.

RN = 17

Radii of a are .

Step 2 Use the Pythagorean Theorem.

SN2 + RS2 = RN2

SN2 + 82 = 172

Substitute 8 for RS and 17 for RN.

SN2 = 225

Subtract 82 from both sides.

SN= 15

Take the square root of both sides.

Step 3 Find NP.

NP= 2(15) = 30


11 2

Step 1 Draw radius PQ.

PS QR , so PS bisects QR.

Check It Out! Example 4

Find QR to the nearest tenth.

PQ = 20

Radii of a are .

Step 2 Use the Pythagorean Theorem.

TQ2 + PT2 = PQ2

TQ2 + 102 = 202

Substitute 10 for PT and 20 for PQ.

TQ2 = 300

Subtract 102from both sides.

TQ 17.3

Take the square root of both sides.

Step 3 Find QR.

QR= 2(17.3) = 34.6


11 2

Lesson Quiz: Part I

1. The circle graph shows the types of cuisine available in a city. Find mTRQ.

158.4


11 2

Lesson Quiz: Part II

Find each measure.

2. NGH

139

3. HL

21


11 2

Lesson Quiz: Part III

4. T U, and AC = 47.2. Find PL to the nearest tenth.

 12.9


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