Question

If a line intersects two concentric circles (circles with the same center) with center O at A, B, C and D. Prove that AB=CD

Answer 1

$\mathfrak{Question:-}$

If a line intersects two concentric circles (circles with the same center) with center O at A, B, C and D. Prove that AB=CD.

$\mathfrak{Answer:-}$

$\small\textsf{Note:-}$ Diagram refers to attachment.

Let AD and BC be the chord of bigger and small circles.

By the definition of segments, we have

AP = AB + BP     ................(1)

and

PD = CP + CD    ................(2)

Let OP be the perpendicular line to the chords.

Since the perpendicular line to the chord bisects the chord,

OP bisects the chord AD and OP bisects the chord BC.

Thus,

AP = PD     .............(3)

and

BP = CP     .............(4)

Substract equation (3) from equation (4)

AP - BP = PD - CP

AB = CD

(Hence Proved)