Page 162 - Math Course 3 (Book 2)

Circles: Arcs and Chords

MO. 11 - L3b Since radius WL is perpendicular to chord HK,
HL ≅ LK.
The Relationships Between

Chords and Diameters mMK + mKL = mMKL Arc addition postulate

mMK + mMHL = mMKL Substitution
THEOREM 11.3

mMK + 53 = 180 Substitution
In a circle, if a diameter (or radius) is perpendicular
to a chord, then it bisects the chord and its arc. mMK = 127 Subtract 53
from each side.
Example:
Answer mMK ≈ 127
If BA ⟂ TV, then UT ≅ UV and AT ≅ AV.
B
Circle W has a radius of 10 centimeter, Radius WL is
is perpendicular to chord HK, which is 16 centime-
ters long. Find JL.
C M

U
T V

A W
J
H K
THEOREM 11.4 L

In a circle or in congruent circles, two chords are Draw radius WK. △WJK is a right triangle.
congruent if and only if they are equidistant from
the center.
WK = 10 r = 10

Let’s Begin ML bisects HK A radius perpendicular to a
chord bisects it.

1
JK = (HK) Definition of segment bisector
2
Radius Perpendicular to a Chord 1
= (16) or 8 HK = 16
2
Example
Use the Pythagorean Theorem to find WJ.
Circle W has a radius of 10 centimeter, Radius WL
is perpendicular to chord HK, which is 16 centime- (WJ)² + (JK)² = (WK)² Pythagorean Theorem
ters long. If mHL ≈ 53, find mMK. (WJ)² + 8² = 10² JK = 8, WK = 10
M
(WJ)² + 64 = 100 Simplify.
Subtract 64 from each
(WJ)² = 36
W side.
J
H K WJ = 6 Take the square root of
each side.
L
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