Bell work Bell work Find the value of radius, x, if the Find the value of radius, x, if the diameter of a circle is 25 ft.diameter of a circle is 25 ft.

25 ft

x

Bell work AnswerBell work Answer

Radius, x, is 12.5 ftRadius, x, is 12.5 ft

Unit 3 : Circles: Unit 3 : Circles: 10.2 Arcs and Chords10.2 Arcs and Chords

Objectives: Students will:Objectives: Students will:

1. Use properties of arcs and chords to 1. Use properties of arcs and chords to

solve problems related to circles.solve problems related to circles.

Words for CirclesWords for Circles

1.1. Central AngleCentral Angle

2.2. Minor ArcMinor Arc

3.3. Major ArcMajor Arc

4.4. SemicircleSemicircle

5.5. Congruent ArcsCongruent Arcs

6.6. ChordChord

7.7. Congruent ChordsCongruent Chords

Check your answers to see how you did.

Are there any words/terms that you are unsure of?

Label Circle PartsLabel Circle Parts

1.1. SemicirclesSemicircles2.2. CenterCenter3.3. DiameterDiameter4.4. RadiusRadius

9. Tangent9. Tangent10. Secant10. Secant11. Minor Arc11. Minor Arc12. Major Arc12. Major Arc

5. Exterior6. Interior7. Diameter8. Chord

Arcs of Circles Arcs of Circles CENTRAL ANGLECENTRAL ANGLE – An angle with its – An angle with its vertex at the center of the circlevertex at the center of the circle

Central Angle

60º•

CENTER P

P

A

B

•

•

Central Angle

60º•

CENTER P

P

A

B

•

•

Arcs of Circles Arcs of Circles

MINOR ARC

AB

C•MAJOR ARC

ACB

Minor Arc ABMinor Arc AB and and Major Arc ACBMajor Arc ACB

Arcs of Circles Arcs of Circles

Central Angle

60º•

CENTER P

P

A

B

•

•

Measure of theMINOR ARC = the measure of theCentral Angle

AB = 60ºC •

The measure of the MAJOR ARC = 360 – the measure ofthe MINOR ARC ACB = 360º - 60º = 300º

TheThe measure of themeasure of the Minor Arc AB Minor Arc AB = the measure of the Central = the measure of the Central AngleAngle The measure of the The measure of the Major Arc ACB Major Arc ACB = 360= 360º - the measure of the º - the measure of the Central AngleCentral Angle

300º

Arcs of Circles Arcs of Circles

SemicircleSemicircle – an arc whose endpoints – an arc whose endpoints

are also the endpoints of the diameter are also the endpoints of the diameter

of the circle; of the circle; Semicircle Semicircle = 180= 180ºº

•

180º

Semicircle

Arc Addition PostulateArc Addition Postulate

The measure of an arc formed by two The measure of an arc formed by two adjacent arcs is the sum of the adjacent arcs is the sum of the measures of the two arcsmeasures of the two arcs

AB + BC = ABC

170º + 8 0º = 2 5 0º

•

••A

C

170º 80º

ARC ABC = 250º•

B

Example 1Example 1

••

••••X

Y

P

••Z

75º

110°

Arc Addition PostulateArc Addition Postulate

Answers:Answers:

75110185175

arcXYarcYZarcXYZarcXZ

(p. 605) Theorem 10.4(p. 605) Theorem 10.4

In the same circle or in congruent circles In the same circle or in congruent circles two minor arcs are congruent iff their two minor arcs are congruent iff their corresponding chords are congruentcorresponding chords are congruent

Congruent Arcs and Chords Congruent Arcs and Chords TheoremTheorem

Example 1: Given that Chords DE is Example 1: Given that Chords DE is

congruent to Chord FG. Find the value congruent to Chord FG. Find the value

of x.of x. Arc DE = 100º

Arc FG = (3x +4)º

D E

F G

Congruent Arcs and Chords Congruent Arcs and Chords TheoremTheorem

Answer: x = 32Answer: x = 32ºº

Arc DE = 100º

Arc FG = (3x +4)º

D E

F G

Congruent Arcs and Chords Congruent Arcs and Chords TheoremTheorem

Example 2: Given that Arc DE is Example 2: Given that Arc DE is

congruent to Arc FG. Find the value congruent to Arc FG. Find the value

of x.of x.

Chord DE = 25 in

Chord FG = (3x + 4) in

D E

F G

Congruent Arcs and Chords Congruent Arcs and Chords TheoremTheorem

Answer: x = 7 inAnswer: x = 7 in

Chord DE = 25 in

Chord FG = (3x + 4) in

D E

F G

(p. 605) Theorem 10.5(p. 605) Theorem 10.5

If a diameter of a circle is perpendicular to a chord, If a diameter of a circle is perpendicular to a chord, then the diameter bisects the chords and its arcs. then the diameter bisects the chords and its arcs.

••

Diameter

Chord

PCongruent Arcs

Congruent Segments

(p. 605) Theorem 10.6(p. 605) Theorem 10.6

If one chord is the perpendicular bisector of another If one chord is the perpendicular bisector of another chord then the first chord is the diameter chord then the first chord is the diameter

••

Chord 1: _|_ bisectorof Chord 2, Chord 1 =the diameter

Chord 2

P

Diameter

(p. 606) Theorem 10.7(p. 606) Theorem 10.7In the same circle or in congruent circles, two In the same circle or in congruent circles, two chords are congruent iff they are equidistant from chords are congruent iff they are equidistant from the center. the center. (Equidistant means same perpendicular (Equidistant means same perpendicular distance)distance)

Chord TS Chord QR __ __iff PU VU

••P

Q

R

S

T

U

V

Center P

Example Example

Find the value of Chord QR, if TS = 20 Find the value of Chord QR, if TS = 20

inches and PV = PU = 8 inchesinches and PV = PU = 8 inches

••P

Q

R

S

T

U

V

8 in

8 in

Center P

AnswerAnswer

Chord QR = 20 inches Chord QR = 20 inches

(Theroem 10.7)(Theroem 10.7)

Home workHome work

PWS 10.2 APWS 10.2 A

P. 607- 608 (12-46) evenP. 607- 608 (12-46) even

JournalJournal

Write two things about Arcs and Write two things about Arcs and

Chords related to circles from this Chords related to circles from this

lesson.lesson.