Two concentric circles of radii 3 cm and 5 cm are given. Then length of chord BC which touches the inner circle at P is equal to

A
8cm
B
9 cm
C
10 cm
D
7

Solution:

Given that the radius of the inner circle is 3 cm and that of the outer circle is 5 cm.
We have to find the length of the chord BC.
We know that,
Property 1: Tangents drawn from an external point to a circle are equal.
Property 2: Tangent at any point of the circle is perpendicular to the radius of the circle.
Given that

O Q ⊥ A B

.
Applying the Pythagoras theorem in the

ΔAOQ

, we get,

A O 2 = A Q 2 + O Q 2 ⇒ A Q 2 = 52 − 32 ⇒ A Q 2 = 25 − 9 ⇒ A Q 2 = 16 ⇒ A Q = 4

As OQ bisects AB,

A Q = Q B = 4 c m

.
By property 1, we get that

BQ = BP = 4 cm .

By property 2,

O P ⊥ B C ⇒ B P = P C

Thus,

PC = 4 cm

.
We know that

BC = BP + PC

⇒ B C = 4 + 4 ⇒ B C = 8 c m

Hence, the length of the chord BC is 8 cm.
Therefore, the correct option is 1.

Solution:

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