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

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 center 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

So, the distance of the chord from the center 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 center.

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

So, 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 center 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 + 5²

OB²  =  576 + 25

OB²  =  601

OB = √601

OB = 24.5

So, the radius of the circle is 24.5 cm.

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