State true or false.AB and AC are two chords of a circle of radius r such that AB = 2AC. If p and q are the distances of AB and AC from the center, then 4 q 2 = p 2 +2 r 2 .

Given that,

A B = 2 A C

and the form the center distance of AB and AC are p and q.

Let AC = a and AB = 2a.
As the perpendicular drawn to the constructed-on AC and AB from center O at M and N, thus:

A M = M C = a 2 A N = N B = a

Δ O A M

, using Pythagorean theorem,

A O ² = A M ² + M O ² A O ² = a 2 2 + q ² 1

Δ O A N

, using Pythagorean theorem,

A O ² = A N ² + N O ² A O ² = a ² + p ² (2)

Δ O A N

, using Pythagorean theorem,

r ² = a ² + p ² 3

Simplifying,

a 2 2 + q ² = a ² + p ² a 2 4 + q ² = a ² + p ² 4 q ² = 3 a ² + 4 p ² 4 q ² = p ² + 3 a ² + 3 p ² 4 q ² = p ² + 3 a ² + p ²

Using equation (3) we have

4 q ² = p ² + 3 r ²

.
So, the statement is false, and if p and q are the distances of AB and AC from the center, then

4 q ² = p ² + 3 r ²

.
Hence, option 2 is correct.

State true or false.

AB and AC are two chords of a circle of radius r such that AB = 2AC. If p and q are the distances of AB and AC from the center, then $4 q 2 = p 2 + 2 r 2$ .