Transcript Chapter 12 Section 6 (Lines and Segments in Circles)

DRILL
E
• Solve for x:
220°
D
F
(2x+ 5)°
100°
G
H
12.6
Segment Lengths in Circles
Geometry
Mr. Calise
Objectives/Assignment
• Find the lengths of segments of
chords.
• Find the lengths of segments of
tangents and secants.
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
C
E
D
A
Theorem 12.14
B
• If two chords intersect
in the interior of a circle,
then the product of the C
E
lengths of the segments
of one chord is equal to
A
the product of the
lengths of the segments
of the other chord.
EA • EB = EC • ED
D
Proving Theorem 12.14
B
• You can use similar
triangles to prove Theorem
10.15.
• Given: AB , CD are chords
that intersect at E.
• Prove: EA • EB = EC • ED
C
E
A
D
Proving Theorem 12.14
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
S
• Chords ST and PQ
intersect inside the
circle. Find the value
of x.
Q
9
3
R
X
6
T
RQ • RP = RS • RT
Use Theorem 10.15
9•x=3•6
Substitute values.
Simplify.
9x = 18
x=2
Divide each side by 9.
P
S
Q
92x
12
3
R
X
10
15
6
T
P
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.
R
Q
P
S
Theorem 12.16
• 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.
A
E
C
D
(EA)2 = EC • ED
10
6
x
8
12
(BA)2 = BC • BD
(5)2 = x • (x + 4)
25 = x2 + 4x
0 = x2 + 4x - 25
x=
 4  4 2  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.
Theorem 12.15
B
A
• If two secant segments
E
share the same endpoint
C
outside a circle, then the
D
product of the length of
one secant segment and
the length of its external
segment equals the
EA • EB = EC • ED
product of the length of
the other secant segment
and the length of its
external segment.
Finding Segment Lengths
• Find the value of x.
P
Q
11
9
R
10
S
x
T
RP • RQ = RS • RT
Use Theorem 10.16
9•(11 + 9)=10•(x + 10) Substitute values.
180 = 10x + 100
80 = 10x
8 =x
Simplify.
Subtract 100 from each side.
Divide each side by 10.
10
6
A
x
C
B
E
15
D
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.
Congratulations
on Finishing the Geometry Curriculum
• Chapter 12 Test on Wednesday!
• REVIEW Days June 3rd, 4th, 7th, 8th, 9th, 10th,
11th, 14th (PD 3 Only), 15th (PD 3 Only)
• FINAL EXAM
• PD 2: Tuesday June 15th (7:55 – 9:30)
• PD 3: Tuesday June 15th (12:30 – 2:05)