Transcript 10.5 Segment Lengths in Circles Geometry Mrs. Spitz Spring 2005 Objectives/Assignment • Find the lengths of segments of chords. • Find the lengths of segments of tangents and.

10.5 Segment
Lengths in Circles
Geometry
Mrs. Spitz
Spring 2005
Objectives/Assignment
• Find the lengths of segments of
chords.
• Find the lengths of segments of
tangents and secants.
• Assignment: pp. 632-633 #1-30 all.
• Practice Quiz pg. 635 #1-7 all.
Finding the Lengths of Chords
• When two chords intersect in the
interior of a circle, each chord is
divided into two segments which are
called segments of a chord. The
following theorem gives a relationship
between the lengths of the four
segments that are formed.
B
Theorem 10.15
• 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.
C
E
D
A
EA • EB = EC • ED
Proving Theorem 10.15
• You can use similar
triangles to prove Theorem
10.15.
• Given: AB , CD are chords
that intersect at E.
• Prove: EA • EB = EC • ED
B
C
E
A
D
Proving Theorem 10.15
Paragraph proof: Draw DB
and AC . Because C and
B intercept the same arc,
C  B. Likewise, A 
D. By the AA Similarity
Postulate, ∆AEC  ∆DEB.
So the lengths of
corresponding sides are
proportional.
EA
ED
=
EC
EB
EA • EB = EC • ED
B
C
E
A
Lengths of sides are
proportional.
Cross Product Property
D
Ex. 1: Finding Segment
Lengths
• Chords ST and PQ
intersect inside the
circle. Find the
value of x.
Q
S
9
R
X
6
T
RQ • RP = RS • RT
Use Theorem 10.15
9•x=3•6
Substitute values.
Simplify.
9x = 18
x=2
3
Divide each side by 9.
P
Using Segments of Tangents
and Secants
• In the figure shown,
PS is called a
P
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.
R
Q
S
Theorem 10.16
• If two secant
A
segments share the
same endpoint outside E
C
a circle, then the
product of the length
of one secant
segment and the
EA • EB = EC • ED
length of its external
segment equals the
product of the length
of the other secant
segment and the
length of its external
segment.
B
D
Theorem 10.17
A
• If a secant segment
and a tangent
E
segment share an
C
endpoint outside a
circle, then the
product of the length
of the secant segment (EA)2 = EC • ED
and the length of its
external segment
equal the square of
the length of the
tangent segment.
D
Ex. 2: Finding Segment
Lengths
P
Q
11
9
• Find the value of x.
R
10
S
x
RP • RQ = RS • RT
Use Theorem 10.16
9•(11 + 9)=10•(x + 10) Substitute values.
Simplify.
180 = 10x + 100
Subtract 100 from each side.
80 = 10x
Divide each side by 10.
8 =x
T
Note:
• In Lesson 10.1, you learned how to
use the Pythagorean Theorem to
estimate the radius of a grain silo.
Example 3 shows you another way to
estimate the radius of a circular
object.
Ex. 3: 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.
(CB)2 = CE • CD
(20)2  8 • (2r + 8)
400  16r + 64
336  16r
21  r
Use Theorem 10.17
Substitute values.
Simplify.
Subtract 64 from each side.
Divide each side by 16.
So, the radius of the tank is about 21 feet.
(BA)2 = BC • BD
(5)2 = x • (x + 4)
25 = x2 + 4x
0 = x2 + 4x - 25
x=
 4  42  4(1)(25)
2
x=
 2  29
Use Theorem 10.17
Substitute values.
Simplify.
Write in standard form.
Use Quadratic Formula.
Simplify.
Use the positive solution because lengths cannot be
negative. So, x = -2 + 29  3.39.
Reminders:
• Quiz after this section either
Thursday or Friday.
• Test will be after 10.7 next week
probably Tuesday or Wednesday.
• Chapter 10 Algebra Review can be
done for Extra Credit. Show all
work!!!