Q1.prove that maximum area of right triangle inscribed in a circle is isosceles

Given a circle with radius r.

Let triangle ABC be a right triangle inscribed in the circle.

Since the triangle is a right triangle, hence one of the angle is 90^o.

Let angle C = 90^o

Then AB is diameter of the circle, since angle in semi circle is 90^o

Hence, AB= 2r, let BC=x and AC = y

Area of traingle ABC = A = 1/2 * x*y

Applying pythagoras theorem in triangle ABC, we het

y = ((2r)²-x²) = (4r²-x²)

Putting this value of y in area

A = 1/2 * x * (4r²-x²)

Differentiating A with respect to x, we get