# APPLYING PROPERTIES OF CHORD EXAMPLES

## Properties of Chords of in a Circle

Property 1 : Equal chords of a circle subtend equal angles at the center.

AOB  =  DOC

Property 2 : Perpendicular from the center of a circle to a chord bisects the chord.

AC  =  BC

Property 3 : Equal chords of a circle are equidistant from the center.

OM  =  OL

Property 4 : If two chords intersect inside a circle, then the measure of each angle formed is one half the sum of the measures of the arcs intercepted by the angle and its vertical angle.

m1  =  (1/2) (Measure arcCD+measure of arcAB)

m2  =  (1/2) (Measure arcBC+measure of arcCD)

Property 5 : If two chords intersect inside a circle, then 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.

EA  EB  =  EC  ED

## Examples

Example 1 :

Find the values of x in the diagram shown below. Solution :

Equal chords of a circle subtend equal angles at the center.

In the above diagram,

ACE  =  BCD

Then

AE  =  BD

2x - 5  =  x

Subtract x from each side.

x - 5  =  0

x  =  5

Example 2 :

Find the values of x in the diagram shown below. If two chords intersect inside a circle, then the measure of each angle formed is one half the sum of the measures of the arcs intercepted by the angle and its vertical angle.

x  =  (1/2) [measure of arc AB + measure of arc DC]

x  =  (1/2) [80 + 40]

x  =  (1/2)(120)

x  =  60

Example 3 :

Find the values of x in the diagram shown below. Perpendicular from the centre of a circle to a chord bisects the chord.

AC  =  BC  =  4 cm

In triangle OCB,

OB2  =  OC2 + BC2

x2  =  32 + 42

x2  =  9 + 16

x2  =  25

x  =  √25

x  =  5 cm

Example 4 :

Find the values of x in the diagram shown below. Solution :

EH ⋅ HG  =  JH ⋅ HF

⋅ 10  =  8 ⋅ x

x  =  (⋅ 10) / 8

x  =  40/8

x  =  5

Example 5 :

AB is a diameter of the circle below. If BC = 2 m and AB = 9 m, find the exact length of AC . Solution :

In triangle ABC,

BCA  =  90°

By Pythagorean theorem,

AB2  =  AC2 + BC2

92  =  AC2 + 22

81  =  AC+ 4

Subtract 4 from each side.

77  =  AC2

77  =  AC2

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