Ab Is Diameter Of Circle And C Is Any Point On Circle

AB is a diameter of a circle and C is any point on the circle. Show that the area of △ABC is maximum, when it is isosceles.

Answers:

We have, AB = 2r
And ∠ACB = 90°
Let ∠ABC = θ
⇒ AC = 2r sin θ and BC = 2r cos θ 
∴ Area of △ABC, △ = 1/2(2r sin θ)(2r cos θ)
= r2 sin 2 θ 
Clearly, △ is maximum when sin 2θ is maximum.
∴ △max = r2, when sin 2θ = 1 
or 2θ = π/2 ⇒ θ = π/4
⇒ Triangle is isosceles.