10.2 Arcs and Chords - Montgomery County Schools Arc… · Geometry Using Arcs of Circles • in...

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10.2 Arcs and Chords

Geometry

Mr. Peebles

Spring 2013

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Bell Ringer: Solve For r .

8 ft.

16 ft.

r

r

A

B

C

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Bell Ringer

Answer

(r + 8)2 = r2 + 162

Pythagorean Thm.

Substitute values

c2 = a2 + b2

r 2 + 16r + 64 = r2 + 256 Square of binomial

16r + 64 = 256

16r = 192

r = 12

Subtract r2 from each side.

Subtract 64 from each side.

Divide.

The radius of the silo is 12 feet.

8 ft.

16 ft.

r

r

A

B

C

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Daily Learning Target (DLT)

Monday May 13, 2013

• “I can use properties of arcs and chords

of circles, as applied.”

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Assignment: • Pgs. 577-580 (1-27 Odds, 41-43)

1. 9π m2 19. 3.3 m2

3. 0.7225π ft2 21. 120.4 cm2

5. 86,394 ft2 23. (54π + 20.25 ) cm2

7. 40.5π yds2 25. (4 – π) ft2

9. 169π/6 m2 27. (784 – 196π) in2

11. 12π ft2 41. A

13. 25π/4 in2 42. G

15. 24π in2 43. 5.2 m2

17. 22.1 cm2

3

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Using Arcs of Circles

• In a plane, an angle

whose vertex is the

center of a circle is

a central angle of

the circle. If the

measure of a

central angle, APB

is less than 180°,

then A and B and the

points of P

central angle

minorarcmajor

arcP

B

A

C

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Using Arcs of Circles

• in the interior of APB

form a minor arc of the

circle. The points A and

B and the points of P

in the exterior of APB

form a major arc of the

circle. If the endpoints

of an arc are the

endpoints of a diameter,

then the arc is a

semicircle.

central angle

minorarcmajor

arcP

B

A

C

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Naming Arcs

• For instance, m =

mGHF = 60°.

• m is read “the

measure of arc GF.”

You can write the

measure of an arc next

to the arc. The

measure of a

semicircle is always

180°.

EH F

G

E

60°

60°

180°

GF

GF

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Naming Arcs

• The measure of a

major arc is defined as

the difference between

360° and the measure

of its associated minor

arc. For example, m

= 360° - 60° = 300°.

The measure of the

whole circle is 360°.

EH F

G

E

60°

60°

180°

GEF

GF

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Ex. 1: Finding Measures of Arcs

• Find the measure

of each arc of R.

a.

b.

c.

MN

MPN

PMN PR

M

N80°

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of each arc of R.

a.

b.

c.

Solution:

is a minor arc, so

m = mMRN

= 80°

MN

MPN

PMN PR

M

N80°

MN

MN

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of each arc of R.

a.

b.

c.

Solution:

is a major arc, so

m = 360° – 80°

= 280°

MN

MPN

PMN PR

M

N80°

MPN

MPN

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of each arc of R.

a.

b.

c.

Solution:

is a semicircle, so

m = 180°

MN

MPN

PMN PR

M

N80°

PMN

PMN

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Note:

• Two arcs of the same circle are adjacent if they intersect at exactly one point. You can add the measures of adjacent areas.

• Postulate 26—Arc Addition Postulate. The measure of an arc formed by two adjacent arcs is the sum of the measures of the two arcs.

ABC

B

A

C

m = m + m ABBC

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• Find the measure of

each arc.

a.

b.

c.

m = m + m =

40° + 80° = 120°

GE

R

EF

G

H

GEFGFGE

GH

HE

40°

80°

110°

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each arc.

a.

b.

c.

m = m + m =

120° + 110° = 230°

GE

R

EF

G

H

GEFGF

EF

40°

80°

110° GEF

GE

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each arc.

a.

b.

c.

m = 360° - m =

360° - 230° = 130°

GE

R

EF

G

H

GEFGF

40°

80°

110° GF

GEF

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Ex. 3: Identifying Congruent Arcs

• Find the measures

of the blue arcs.

Are the arcs

congruent?

C

D

A

B45°

45°

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of the blue arcs.

Are the arcs

congruent?

C

D

A

BAB • and are in

the same circle and

m = m = 45°.

So,

DC

AB

DCDC

AB

45°

45°

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Q

S

P

R



of the blue arcs.

Are the arcs

congruent? 80°

80°

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Q

S

P

R



of the blue arcs.

Are the arcs

congruent? RS

PQ • and are in

congruent circles and

m = m = 80°.

So,

PQ

RSRS

PQ

80°

80°

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X

W

Y

Z


• Find the measures of the blue arcs. Are the arcs congruent?

65°

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X

W

Y

Z

• m = m = 65°, but

and are not arcs of the

same circle or of

congruent circles, so

and are NOT

congruent.


• Find the measures of the blue arcs. Are the arcs congruent?

65° XY

ZW

XY

ZW

XY

ZW

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Using Chords of Circles

• A point Y is called

the midpoint of

if . Any

line, segment, or ray

that contains Y

bisects . XYZ

XYYZ

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Theorem 10.4

• In the same circle, or in

congruent circles, two

minor arcs are

congruent if and only if

their corresponding

chords are congruent.

AB BC if and only if

AB BC

C

B

A

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Theorem 10.5

• If a diameter of a

circle is

perpendicular to a

chord, then the

diameter bisects the

chord and its arc. E

D

G

F

DG

GF

, DE EF

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Theorem 10.5

• If one chord is a

perpendicular

bisector of another

chord, then the first

chord is a diameter.

JK is a diameter of

the circle.

J

L

K

M

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Ex. 4: Using Theorem 10.4

• You can use

Theorem 10.4 to

find m . AD

BA

C

2x°

(x + 40)°

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Ex. 4: Using Theorem 10.4

• You can use

Theorem 10.4 to

find m . AD

• Because AD DC,

and . So,

m = m

ADDC

ADDC

2x = x + 40 Substitute

x = 40 Subtract x from each

side.

BA

C

2x°

(x + 40)°

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or in congruent

circles, two chords

are congruent if and

only if they are

equidistant from the

center.

• AB CD if and only

if EF EG.

Theorem 10.7

F

G

E

B

A

C

D

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Exit Quiz – 5 Points

• Find the area of the sector shown

below.

4 ft.

P

D

C

Sector CPD intercepts an

arc whose measure is 85°.

The radius is 4 ft.

A m

360° = • r2

10.2 Arcs and Chords - Montgomery County Schools Arc… · Geometry Using Arcs of Circles • in...

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