Arcs and ChordsArcs and Chords

Chapter 10-3

• inscribed

• circumscribed

• Recognize and use relationships between arcs and chords.

• Recognize and use relationships between chords and diameters.

Standard 7.0 Students prove and use theorems involving the properties of parallel lines cut by a transversal, the properties of quadrilaterals, and the properties of circles. (Key)

Standard 21.0 Students prove and solve problems regarding relationships among chords, secants, tangents, inscribed angles, and inscribed and circumscribed polygons of circles. (Key)

Chord TheoremsChord Theorems• In the same circle or circles, 2 minor arcs

are their corresponding chords are

BC ED BC EDB

A

C

D

E

Prove Theorem 10.2

PROOF Write a two-column proof.

Prove:

Given:

is a semicircle.

Prove Theorem 10.2

Proof:Statements Reasons

5. Def. of arc measure5.

4. Def. of arcs4.

2. Def. of semicircle2.

3. In a circle, if 2 chords are , corr. minor arcs are .

3.

Answer:

1. 1. Givenis a semicircle.

Prove Theorem 10.2

Answer:

6. 6. Arc AdditionPostulate

7. 7. Substitution

8. 8. Subtraction Property and simplify

9. 9. Division Property

11. 11. Substitution

Statements Reasons

10. 10. Def. of arc measure

PROOF Choose the best reason to complete the following proof.

Prove:

Given:

Proof:Statements Reasons

1.

2.

3.

4.

1. Given

2. In a circle, 2 minor arcs are , chords are .

3. ______

4. In a circle, 2 chords are , minor arcs are .

A. A

B. B

C. C

D. D

0% 0%0%0%

A. Segment Addition Postulate

B. Definition of

C. Definition of Chord

D. Transitive Property

Inscribed Polygons• If all the vertices of a polygon lie on the circle

– The polygon is inscribed in the circle– The circle is circumscribed about the polygon

A regular hexagon is inscribed in a circle as part of a logo for an advertisement. If opposite vertices are connected by line segments, what is the measure of angle P in degrees?

Since connecting the opposite vertices of a regular hexagon divides the hexagon into six congruent triangles, each central angle will be congruent. The measure of each angle is 360 ÷ 6 or 60.

Answer: 60

1. A

2. B

3. C

0% 0%0%

ADVERTISING A logo for an advertising campaign is

a pentagon that has five congruent central angles.

Determine whether

A. yes

B. no

C. cannot be determined

Chord TheoremsChord Theorems• If the diameter of a circle is to a chord, the

diameter bisects the chord and its arc

AB BC

AD DC A

B

C

D

Radius Perpendicular to a Chord

Since radius is perpendicular to chord

Arc addition

Substitution

Substitution

Subtraction

Radius Perpendicular to a Chord

A radius perpendicular to a chord bisects it.

Def of seg bisector

8

10

Use the Pythagorean Theorem to find WJ.

Pythagorean Theorem

Simplify.

Subtract 64 from each side.

Take the square root of each side.

JK = 8, WK = 10

8

10

Segment Addition Postulate

Subtract 6 from each side.

WJ = 6, WL = 10

6

1. A

2. B

3. C

4. D

0%0%0%0%

A B C D

A. 35

B. 70

C. 105

D. 145

1. A

2. B

3. C

4. D

0%0%0%0%

A B C D

A. 15

B. 5

C. 10

D. 25

Chord TheoremsChord Theorems• In the same circle or circles, 2 chords are

they are equidistant from the center.

F GE

C

D

A

B

& AB CD EF EG AB CD

Chords Equidistant from Center

15

24

12

Pythagorean Theorem

24

9

A. A

B. B

C. C

D. D

0% 0%0%0%

A. 12

B. 36

C. 72

D. 32

A. A

B. B

C. C

D. D

0% 0%0%0%

A. 12

B. 36

C. 72

D. 32

Chord Theorems Sample Problem• Solve for x + y

B

AB BC

AD DC AD = 3x + 7; DC = 5x +3

m AB = 4y + 8; m AEC = 96

3x + 7 = 5x + 3

4 = 2x

2=x

AB ½ AC

m AC = m AEC

m AC = 964y + 8 = ½ (96)

4y + 8 = 48

4y = 40

y = 10

A

C

D

E

Homework Chapter 10.3

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