93: Arcs and chords

Arc of the chord: Occurs when a minor arc and a
chord have the same endpoints.
P
9­3: Arcs and chords
Theorem: In a circle, if a diameter of a circle is
perpendicular to a chord, then the diameter bisects
C
Q
In circle C, PQ is the arc of PQ.
(
the chord and its arc.
T
C
Q
S
R
Given: CS TR
Then: TQ ≅ RQ
and
RS ≅ TS
(
(
If a chord is a perpendicular bisector of another chord,
then the first chord is a diameter. (converse)
Example 1
The Diameter of a ferris wheel is 56 ft and
seats are connected with 6 ft steel beams.
Find the length of the support bar.
Theorem: Minor arcs are congruent iff their
corresponding chords are congruent.
A
B
56
[
6
C
P
Q
(
(
AB ≅ PQ iff AB ≅ PQ
J
L
(
(
JK ≅ LM iff JK ≅ LM
K
M

Theorem: In the same circle, or in congruent
circles, two chords are congruent iff they are
equidistant from the center.
Y
Example 2
Suppose a chord is 20 inches long and 24
inches from the center. Find the radius.
Z
C
x
y
X
W
YW ≅ ZX iff x = y
Remember: The theorem
states the diameter bisects
the chord if it is perpendicular.
Example 3
A chord is 5 inches from center and 24
inches long. Find the diameter.
Example 4
The diameter is 30 cm long and a chord is 24 cm
long. Find the distance from the chord to the
center of the circle.