Lesson 84 Arcs and Chords 1 WarmUp 35 Wednesday 518 Lesson 84 Arcs and Chords 2 1 Find the perimeter of the polygon 2 Find the distance between the centers of the pulleys 3 The radius of Earth is about 6400 km Find the distance d given h 1km ID: 730216
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Slide1
Warm-Up #34 Tuesday, 5/17
Lesson 8-4: Arcs and Chords
1Slide2
Warm-Up #35 Wednesday, 5/18
Lesson 8-4: Arcs and Chords
2
1. Find the perimeter of the polygon
2. Find the distance between the centers of the pulleys.
3
. The radius of Earth is about 6400 km. Find the distance d, given h= 1kmSlide3
Arcs and Chords page 1 and 2
Lesson 8-4: Arcs and Chords
3Slide4
4
Arcs
and ChordsSlide5
Definition
Central angle
– an angle whose vertex is the center of a circle. Slide6
Definitions
Minor arc
– Part of a circle that measures less than 180
°Major arc – Part of a circle that measures between 180
° and 360°.
Semicircle
– An arc whose endpoints are the endpoints of a diameter of the circle.
Note
: major arcs and semicircles are named with three points and minor arcs are named with two pointsSlide7
Definitions
Measure of a minor arc
– the measure of its central angleMeasure of a major arc – the difference between 360
° and the measure of its associated minor arc.Slide8Slide9
Arc Addition Postulate
The measure of an arc formed by two adjacent arcs is the sum of the measures of the two arcs. Slide10
Example 1
Find the measure of each arc.
70
°
360
° - 70° = 290°
180
°Slide11
Example 2
Find the measures of the red arcs. Are the arcs congruent?Slide12
Example 3
Find the measures of the red arcs. Are the arcs congruent?Slide13
Arcs and Chords Theorem
In the same circle, or in congruent circles, two minor arcs are congruent if and only if their corresponding chords are congruent.Slide14
Perpendicular Diameter Theorem
If a diameter of a circle is perpendicular to a chord, then the diameter bisects the chord and its arc.Slide15
Perpendicular Diameter Converse
If one chord is a perpendicular bisector of another
chord which must pass through the center of the circle, then the first chord is a diameter. Slide16
Congruent Chords Theorem
In the same circle, or in congruent circles, two chords are congruent if and only if they are equidistant from the center
. Slide17
Example 4Slide18
Example 3A: Applying Congruent Angles, Arcs, and Chords
TV
WS
. Find m
WS.
9
n
– 11
=
7
n
+ 11
2
n
= 22
n
= 11
= 88
°
chords have
arcs.
Def. of
arcs
Substitute the given measures.
Subtract 7n and add 11 to both sides.
Divide both sides by 2.
Substitute 11 for n.
Simplify.
m
TV
= m
WS
m
WS
= 7
(11)
+ 11
TV
WSSlide19
Example 3B: Applying Congruent Angles, Arcs, and Chords
C
J
, and
m
GCD
m
NJM.
Find
NM.
GD
=
NM
arcs have
chords.
GD
NM
GD
NM
GCD
NJM
Def. of
chordsSlide20
Example 3B Continued
14
t
– 26
=
5
t
+ 1
9t
= 27
NM
= 5
(3)
+ 1
= 16
Substitute the given measures.
Subtract 5t and add 26 to both sides.
Divide both sides by 9.
Simplify.
t
= 3
Substitute 3 for t.
C
J
, and
m
GCD
m
NJM.
Find
NM.Slide21
Check It Out!
Example 3a
PT
bisects
RPS
. Find
RT
.
6
x
= 20 – 4
x
10
x
= 20
x
= 2
RT
= 6
(2)
RT
= 12
Add 4x to both sides.
Divide both sides by 10.
Substitute 2 for x.
Simplify.
RPT
SPT
RT
=
TS
m
RT
m
TS Slide22
Check It Out!
Example 3b
A
B
,
and
CD
EF
. Find m
CD
.
Find each measure.
25
y
=
(30
y
– 20)
20 = 5
y
4 =
y
CD
= 25
(4)
Subtract 25y from both sides. Add 20 to both sides.
Divide both sides by 5.
Substitute 4 for y.
Simplify.
m
CD
= 100
m
CD
= m
EF
chords have
arcs.
Substitute.Slide23
Find
NP
.
Example 4: Using Radii and Chords
Step 2
Use the Pythagorean Theorem.
Step 3
Find
NP
.
RN
= 17
Radii of a
are
.
SN
2
+
RS
2
=
RN
2
SN
2
+
8
2
=
17
2
SN
2
= 225
SN
= 15
NP
= 2
(15)
=
30
Substitute 8 for RS and 17 for RN.
Subtract 8
2
from both sides.
Take the square root of both sides.
RM
NP , so RM bisects NP.
Step 1
Draw radius
RN
.Slide24
Check It Out!
Example 4
Find QR to the nearest tenth.
Step 2
Use the Pythagorean Theorem.
Step 3
Find
QR
.
PQ
= 20
Radii of a
are
.
TQ
2
+
PT
2
=
PQ
2
TQ
2
+
10
2
=
20
2
TQ
2
= 300
TQ
17.3
QR
= 2
(17.3)
=
34.6
Substitute 10 for PT and 20 for PQ.
Subtract 10
2
from both sides.
Take the square root of both sides.
PS
QR , so PS bisects QR.
Step 1
Draw radius
PQ
.Slide25
Lesson 8-4: Arcs and Chords
25
Try Some Sketches:
Draw a circle with a chord that is 15 inches long and 8 inches from the center of the circle.
Draw a radius so that it forms a right triangle.
How could you find the length of the radius?
8cm
15cm
O
A
B
D
∆ODB is a right triangle and
Solution:
xSlide26
Lesson 8-4: Arcs and Chords
26
Try Some Sketches:
Draw a circle with a diameter that is 20 cm long.
Draw another chord (parallel to the diameter) that is 14cm long.
Find the distance from the smaller chord to the center of the circle.
10 cm
10 cm
20cm
O
A
B
D
C
14 cm
x
E
Solution:
OB (radius) = 10 cm
∆EOB is a right triangle.
7.1 cmSlide27
Lesson Quiz: Part I
1.
The circle graph shows the types of cuisine available in a city. Find mTRQ.
158.4
Slide28
Lesson Quiz: Part II
2.
NGH
139
Find each measure.
3.
HL
21Slide29
Lesson Quiz: Part III
12.9
4.
T
U
, and
AC
= 47.2. Find
PL
to the nearest tenth.