CC Geometry H
Aim 10: What theorem relates to lengths of chords in a circle?
Do Now: Identify the type of angle and the angle/arc relationship. Then find the
measure of x:
B
B
a.
b.
D
x0
F
620
x0
740
A
D
29
C
0
E
C
C
c.
x0
35
d.
D
B
0
x0
A
B
720
E
D
42.50
C
a b = c d
THEOREM: If chords in a circle intersect,
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.
d
a
c
b
Proof of Theorem:
E
B
Draw BD and EC.
Prove: ΔDFB ~ ΔCFE
F
C
D
Statements
Reasons
What is true about similar triangles and corresponding sides?
Write a proportion involving sides BF, FC, DF, and FE. Then rearrange the
proportion to prove the theorem above (BF)(FC) = (DF)(FE)
Exercises
1. Find x.
a)
3
x
b)
4
6
6
12
4
x
2. Chords AB and CD intersect at E. If CD = 13, EB = 3, and CE = 4, find AE.
3. Chords AB and CD intersect at E in a circle. ED is 2 more than CE, BE = 5, EA = 3.
Find CD.
4. DF < FB, DF ≠ 1. DF < FE, and all values are integers. Prove DF = 3.
D
C
F
7
6
B
E
5. If chords AB and RS intersect at point E within a circle so that AE = 3, EB = 16,
and RE:ES = 3:4, find: a) RE b) ES and c) RS.
6. In the circle shown, DE = 11, BC = 10, DF = 8. Find FE, BF, and FC.
E
B
F
C
D
7. In the diagram of circle O below, chord AB intersects chord CD at E,
DE = 2x + 8, EC = 3, AE = 4x – 3, and EB = 4. Find the value of x and the length of
AE.
8. Chords AB and CD intersect at point E within a circle. AE = 9 and EB = 1.
a) If CD = 6 and CE = x, write an expression to represent ED in terms of x.
b) Find CE.
c) Find ED.
9. Find x:
x
x + 4
{
x + 9
Sum it Up
• If chords in a circle intersect, the product of the
lengths of the segments of one chord is equal to the
a
product of the lengths of the segments of the other
chord.
a
b =c
d
d
b
c
Name: ______________________
Date: ____________
Common Core GeometryH
HW #10
1) Find x.
2) Find x:
8
8
x
2
Chords AB and CD
A
C
x
10
2.5
3.
4.
Diameter AB is perpendi-
A
D
O
If CD = 13, EB = 3
E
and CE = 4, find AE.
B
C
A
P
cular to chord CD at E.
B
If AE = 24 and EB = 6,
B
D
D
5.
intersect at E.
E
12
what is CD?
C
Chords AB and CD intersect at P.
If AP = x, PB = y and CP = z, what is the
length of PD, in terms of x, y, and z?
6.
E
A
Chords AB and CD
B
C
intersect at E.
D
7.
C
CE = 5 and ED = 4,
find AE.
x
5
If AE = x, EB = x - 8,
B
Chords AB and CD
A
(1) xy/z
(2) xz/y
(3) yz/x
(4) (x + y)/z
8.
A
Diameter AB is
perpendicular
intersect at E.
3
x+2
D
If AE = 3, EB = 5,
to chord CD at E. If
E
C
CE = x and ED = x + 2,
find the value of x.
B
D
CE = 8, and EB = 4,
find AE.
9. Find x.
10. Find x.
Mixed Review:
12. Find x and y.
11. Find x.
360
3
O
y0
2
820
x
x0
13. Given AE ||BD, find x.
14. Given ΔABC, find the coordinates
the point of intersection of the
medians of ΔABC.