Geometry

Arcs and Chords

April 19, 2023

Goals

Identify arcs & chords in circles Compute arc measures and angle

measures

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Central Angle

An angle whose vertex is the center of a circle.A

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Minor Arc

Part of a circle. The measure of the central angle is less than 180.

A

C

T

CT

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Semicircle

Half of a circle. The endpoints of the arc are the endpoints of a diameter. The central angle measures 180.

A

C

T

CTDD

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Major Arc

Part of a circle. The measure of the central angle is greater than 180.

A

C

T

CTDD

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Major Arc

CTDBUT NOT

A

C

T

D

CDT

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Measuring Arcs An arc has the same measure as the

central angle. We say, “a central angle subtends an arc

of equal measure”.

4242

A

BC

42

42

m ACB

mAB

Central Angle Demo

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Measuring Major Arcs The measure of an major arc is given by

360 measure of minor arc.

42

42

m ACB

mAB42

42A

BCD

360 42 318 mADB

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Arc Addition Postulate

R

ACT

mRAT mRA mAT

Postulate Demonstration

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What have you learned so far? Page 607 Do problems 3 – 8. Answers… 3) 4) 5) 6) 7) 8)

P

Q

R

S

T

120

60

40

60mRS 300mRPS 180mPQR 100mQS 220mQSP

40m QTR

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Subtending Chords

A

BO

C Chord BC subtends BC.

Chord AB subtends AB.

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Theorem 12.4

Two minor arcs are congruent if and only if corresponding chords are congruent.

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Theorem 12.4

AB

CD

If AB CD, then AB CD.

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Example

120 (5x + 10)

Solve for x.

5x + 10 = 120

5x = 110

x = 22

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Theorem 12.5

If a diameter is perpendicular to a chord, then it bisects the chord and the subtended arc.

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Example

52

2x

Solve for x.

2x = 52

x = 26

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Theorem 12.6

If a chord is the perpendicular bisector of another chord, then it is a diameter.

Diameter

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Theorem 12.7

Two chords are congruent if and only if they are equidistant from the center of a circle.

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The red wires are the same length because they are the same distance from the center of the grate.

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Example

16

4x – 2

Solve for x.

4x – 2 = 16

4x = 18

x = 18/4

x = 4.5

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Summary Chords in circles subtend major and

minor arcs. Arcs have the same measure as their

central angles. Congruent chords subtend congruent

arcs and are equidistant from the center.

If a diameter is perpendicular to a chord, then it bisects it.

April 19, 2023

Practice Problems