10.3 Chords and Segment Relationships

* Apply properties of chords
* Find the lengths of segments formed by lines that intersect circles

1. Chords and Segment Relationships
The student is able to (I can):
•Apply properties of chords
•Find the lengths of segments formed by lines that
intersect circles

2. In the same circle (or in congruent circles), two minor arcs
are congruent if and only if their corresponding chords are
congruent.
E
R
O
Fmm
FORE
FORE



3. If a radius or diameter is perpendicular to a chord, then it
bisects the chord and its arc.ERGO⊥
G
E
O
R
AGAAO GRRO

4. Example
Find the length of .BU
•
B
L
U
E
2
5
x

5. Example
Find the length of .BU
•
B
L
U
E
3
2
5222
35x+=
x
x= 4
BU= 2(4) = 8

6. If one chord of a circle is a perpendicular bisector of another
chord, then the first chord is a diameter.
The midpoint of the diameter would be the center of
the circle.
G
W
O
N
A
If is a ⊥bisector
of , then is a
diameter.WN GO WN

7. A while back, I needed to find the center of a circle, so I used
this property. ☺

8. A while back, I needed to find the center of a circle, so I used
this property. ☺
Step 1: I drew
a chord
Step 2: I drew the
perpendicular bisector,
which is a diameter.

9. A while back, I needed to find the center of a circle, so I used
this property. ☺
Step 3: I measured and
found the midpoint of the
diameter, which gave me
the center!

10. chord-chord product –if two chords intersect in the interior
of a circle, then the products of the lengths of the
segments of the chords are equal.
S
P
A
C
EPS EAACAA=

11. Examples
1.Find the value of x.
2.What is the diameter of the circle?
9
12
x6()9612x= 972x= 8x= ()466x= 436x= 9x= diameter4913=+=
6
6
4
x
•

12. secant-secant product –if two secants intersect in the
exterior of a circle, then the product of the lengths of
one secant segment and its external segment equals
the product of the lengths of the other secant
segment and its external segment.
(whole •outside = whole •outside)
O
P
M
E
TMT OMEM MP=

13. Example
Find the value of x.
10
x
8
12( )()10101220
10100240
10140
14
x
x
x
x
+=
+=
=
=

14. secant-tangent product –if a secant and a tangent intersect
in the exterior of a circle, then the product of the
lengths of the secant segment and its external
segment equals the length of the tangent segment
squared.
(whole •outside = tangent
2
)
G
O
R
F2
FOROGO=

15. Example
Find the value of x.
8
10
x( )
()()
2
2
2
1088
188
144
12
x
x
x
x
+=
=
=
=