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Splash Screen. Five-Minute Check (over Lesson 10–2) NGSSS Then/Now Theorem 10.2 Proof: Theorem 10.2 (part 1) Example 1: Real-World Example: Use Congruent Chords to Find Arc Measure Example 2: Use Congruent Arcs to Find Chord Lengths Theorems

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### Splash Screen

Five-Minute Check (over Lesson 10–2)

NGSSS

Then/Now

Theorem 10.2

Proof: Theorem 10.2 (part 1)

Example 1: Real-World Example: Use Congruent Chords to Find Arc Measure

Example 2: Use Congruent Arcs to Find Chord Lengths

Theorems

Example 3: Use a Radius Perpendicular to a Chord

Example 4: Real-World Example: Use a Diameter Perpendicular to a Chord

Theorem 10.5

Example 5: Chords Equidistant from Center

A

B

C

D

A.105

B.114

C.118

D.124

A

B

C

D

A.35

B.55

C.66

D.72

A

B

C

D

A.125

B.130

C.135

D.140

A

B

C

D

A.160

B.150

C.140

D.130

A

B

C

D

A.180

B.190

C.200

D.210

### 5-Minute Check 5

A

B

C

D

Dianne runs around a circular track that has a radius of 55 feet. After running three quarters of the distance around the track, how far has she run?

A.129.6 ft

B.165 ft

C.259.2 ft

D.345.6 ft

### 5-Minute Check 6

MA.912.G.6.2Define and identify: circumference, radius, diameter, arc, arc length, chord, secant, tangent and concentric circles.

MA.912.G.6.4 Determine and use measures of arcs and related angles.

### NGSSS

You used the relationships between arcs and angles to find measures. (Lesson 10–2)

• Recognize and use relationships between arcs and chords.

• Recognize and use relationships between arcs, chords, and diameters.

### Concept

Jewelry A circular piece of jade is hung from a chain by two wires around the stone. JM  KL and = 90. Find .

Use Congruent Chords to Find Arc Measure

### Example 1

Use Congruent Chords to Find Arc Measure

A

B

C

D

A.42.5

B.85

C.127.5

D.170

### Example 1

Use Congruent Arcs to Find Chord Lengths

### Example 2

Use Congruent Arcs to Find Chord Lengths

WX= YZDefinition of congruent segments

7x – 2= 5x + 6Substitution

2x= 8Add 2 to each side.

x= 4Divide each side by 2.

So, WX = 7x – 2 = 7(4) – 2 or 26.

A

B

C

D

A.6

B.8

C.9

D.13

### Concept

Use a Radius Perpendicular to a Chord

A

B

C

D

A.14

B.80

C.160

D.180

### Example 3

CERAMIC TILE In the ceramic stepping stone below, diameter AB is 18 inches long and chord EF is 8 inches long. Find CD.

Use a Diameter Perpendicular to a Chord

### Example 4

Use a Diameter Perpendicular to a Chord

This forms right ΔCDE.

### Example 4

Since diameter AB is perpendicular to EF, AB bisects chord EF by Theorem 10.3. So, DE = (8) or 4 inches.

1

__

2

Use a Diameter Perpendicular to a Chord

Step 2Find CE and DE.

Since AB = 18 inches, CB = 9 inches. All radii of a circle are congruent, soCE = 9 inches.

### Example 4

Use a Diameter Perpendicular to a Chord

Step 3Use the Pythagorean Theorem to find CD.

CD2 + DE2= CE2Pythagorean Theorem

CD2 + 42= 92Substitution

CD2 + 16= 81Simplify.

CD2 = 65Subtract 16 from each side.

Take the positive square root.

### Example 4

A

B

C

D

In the circle below, diameter QS is 14 inches long and chord RT is 10 inches long. Find VU.

A.3.87

B.4.25

C.4.90

D.5.32

### Concept

Since chords EF and GH are congruent, they are equidistant from P. So, PQ = PR.

Chords Equidistant from Center

### Example 5

Chords Equidistant from Center

PQ= PR

4x – 3= 2x + 3Substitution

x= 3Simplify.

So, PQ = 4(3) – 3 or 9

A

B

C

D

A.7

B.10

C.13

D.15