97 Segment Lengths in Circles Objectives Find the lengths of segments of chords Find the lengths of segments of tangents and secants Finding the Lengths of Chords When two chords of a circle intersect ID: 330557
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Slide1
Geometry
9.7
Segment Lengths in CirclesSlide2
Objectives
Find the lengths of segments of chords.
Find the lengths of segments of tangents and secants
.Slide3
Finding the Lengths of Chords
When two chords
of a circle intersect,
each chord is divided into two segments which are called segments of a chord.
There are several possible cases. Slide4
Theorem 9.14
If two chords intersect in the interior of a circle, then the product of the lengths of the segments of one chord is equal to the product of the lengths of the segments of the other chord.
EA
• EB = EC • EDSlide5
Finding Segment Lengths
Chords ST and PQ intersect inside the circle. Find the value of x.
RQ
• RP =
RS
•
RT
Use Theorem
9.14
Substitute values.
9
• x =
3
•
6
9
x = 18
x = 2
Simplify.
Divide each side by 9.Slide6
Using Segments of Tangents and Secants
In the figure shown, PS is called a
tangent segment
because it is tangent to the circle at an end point. Similarly, PR is a
secant segment
and PQ is the
external segment of PR.Slide7
Theorem 9.15
If two secant segments share the same endpoint outside a circle
,
then the product of the length of one secant segment and the length of its external segment equals the product of the length of the other secant segment and the length of its external segment.
EA
• EB =
EC
• EDSlide8
Theorem 9.16
If a secant segment and a tangent segment share an endpoint outside a circle, then the product of the length of the secant segment and the length of its external segment equal the square of the length of the tangent segment.
(
EA
)
2
=
EC
• EDSlide9
Finding Segment Lengths
Find the value of x.
RP
• RQ = RS • RT
Use Theorem 10.16
Substitute values.
9
•(11 + 9)=10•(x + 10)
180
= 10x + 100
80 = 10x
Simplify.
Subtract 100 from each side.
8 = x
Divide each side by 10.Slide10
Note:
In Lesson 10.1, you learned how to use the Pythagorean Theorem to estimate the radius of a
circle.
Example 3 shows you another way to estimate the radius of a circular object.Slide11
Estimating the radius of a circle
Aquarium Tank. You are standing at point C, about 8 feet from a circular aquarium tank. The distance from you to a point of tangency is about 20 feet. Estimate the radius of the tank.Slide12
(
CB
)
2
=
CE
• CD
Use Theorem 10.17
Substitute values.
400
16r + 64
336
16r
Simplify.
21
r
Divide each side by 16.
(20)2
8 • (2r + 8)
Subtract 64 from each side.
So
, the radius of the tank is about 21 feet.Slide13
Practice
3
•
( x + 3)
=
4 (4 + 7)
3x + 9 = 4 * 11
3x = 44- 9
3 x = 35
11.7
3
• 6
= 9
•
x
18 = 9x
2
(
x)2 = 9
* (4+ 9)X2 = 117
X= 10.8Slide14
Practice
2 x = 25
X = 12.5
31
2
= 20( 20 + x )
961 = 400 + 20x
= 20 x
X = 28.1
10 (10 +3x)= 8 (8+ 22)
100+ 30x = 240
30x = 140
x = 4.7Slide15
Practice
6 x = 3 * 8
6x = 24
X = 4
8( 8+ 8 ) = 6 * ( 6 + 2x)
8 * 16 = 36 + 12 x
128 = 36 + 12x
92 = 12x
X = 7.7Slide16
Take Notes
4 * ( 4 + 6) = 5 ( 5 + x)
4 *10 = 25 + 5x
40 = 25 + 5x
15 = 5x
X= 3
Take a minute and solve …
7 ( 7+5) = 6( 6+x)
84 = 36 + 6x
48 = 6x
X = 8Slide17
Take Notes
4 * 9 = 6 x
36 = 6x
X = 6
Take a minute and solve …
18
2
= 12( 12+ x)
324 = 144 +12x
180 = 12x
15Slide18
Take Notes
9 ( x+ 7 + 9) = 8 ( 8 + x + 10)
9 ( x + 16) = 8 ( x + 18)
9x + 144 = 8x + 144
X = 0
Take a minute and solve …
9 ( 9 + x + 4) = 8 ( 8 + x + 8)
9 ( x + 13) = 8 ( x + 16)
9x + 117=8x +128
X = 11Slide19
Take Notes
4 ( 9 + x ) = 6 ( x + 6 )
36 + 4x = 6x + 36
X = 0
Take a minute and solve …
8( 2x + 1) = 6 (3x)
16x + 8 = 18x
2x = 8
X = 4Slide20
Take Notes
40
2
= 32( 32 + x)
1600= 1024 + 32x
576 = 32x
x = 18
Take a minute and solve …
15
2
= 9( 9+x)
225 = 81 + 9x
144 = 9x
16