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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

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

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.

Also addresses MA.912.G.6.3.

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.

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

Use Congruent Chords to Find Arc Measure

A

B

C

D

A.42.5

B.85

C.127.5

D.170

Use Congruent Arcs to Find Chord Lengths

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.

Answer:WX = 26

A

B

C

D

A.6

B.8

C.9

D.13

Answer:

Use a Radius Perpendicular to a Chord

A

B

C

D

A.14

B.80

C.160

D.180

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

Step 1Draw radius CE.

Use a Diameter Perpendicular to a Chord

This forms right ΔCDE.

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.

Answer:

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.

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

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

Chords Equidistant from Center

Chords Equidistant from Center

PQ= PR

4x – 3= 2x + 3Substitution

x= 3Simplify.

So, PQ = 4(3) – 3 or 9

Answer:PQ = 9

A

B

C

D

A.7

B.10

C.13

D.15