Question

In the figure below, AB is a chord and CD is the diameter perpendicular to it. Prove that ABC is an isosceles triangle.

Answer 1

Given: AB is a chord and CD is the diametre of the circle which is perpendicular to AB.

To prove: ΔABC is an isosceles triangle.

Proof:

As CE is perpendicular to AB, ∠AEC = ∠BEC = 90°.

We know that perpendicular from the centre of the circle to the chord bisects the chord.

⇒ AE = EB

Now, in ΔACE and ΔBCE:

AE = EB (Proved above)

∠AEC = ∠BEC = 90° (Proved above)

CE = CE (Common)

∴ ΔACE ≅ ΔBCE (By side angle side criterion)

⇒ AC = BC (By c.p.c.t.)

Thus, ΔABC is an isosceles triangle.