HOW TO FIND THE DISTANCE OF A CHORD FROM THE CENTER OF A CIRCLE

About "How to find the distance of a chord from the center of a circle"

How to find the distance of a chord from the center of a circle ?

Here we are going to see how to find the distance of a chord from the center of a circle.

To find the length of chord, we may use the following theorem

  • Perpendicular from the centre of a circle to a chord bisects the chord.

Example 1 :

A chord of length 16 cm is drawn in a circle of radius 10 cm. Find the distance of the chord from the centre of the circle.

Solution :

AB is a chord of length 16 cm

C is the midpoint of AB.

OB is the radius of length 10 cm

AB = 16 cm

AC = (1/2) ⋅ 16 = 8 cm

OB = 10 cm

In a right triangle OAC.

OC2 = OA2 - AC2

  =   √(102 - 82)

  =   √(100 - 64)

  =   √36 cm

OC  =  6 cm

Hence, the distance of the chord from the centre is 6 cm

Example 2 :

The radius of a circle is 15 cm and the length of one of its chord is 18 cm. Find the distance of the chord from the centre.

Solution :

AB is a chord of length 18 cm

C is the midpoint of AB.

OB is the radius of length 10 cm

AB  =  18 cm

AC  =  (1/2) ⋅ 18  =  9 cm

OB = 15 cm

In a right triangle OCB.

OC2 = OB2 - BC2

  =   √(152 - 92)

  =   √(225 - 81)

  =   √144

OC  =  12 cm

Hence the distance of chord from the center is 12 cm.

Example 3 :

A chord of length 20 cm is drawn at a distance of 24 cm from the centre of a circle. Find the radius of the circle.

Solution :

Here the line OC is perpendicular to AB, which divides the chord of equal lengths.

In Δ OCB,

OB2  =  OC2 + BC2

OB2  =  242 + 102

BC2  =  576 + 100

BC2  =  676

BC  =  √676

BC  =  √(26 ⋅ 26)

BC  =  26  cm

Hence the radius of the circle is 26 cm.

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