show that the right triangle of maximum area that can be inscribed in a circle is an isosceles triangle

Dear Student,

If we consider a circle with center O and triangle ABC inscribed on it If Angle BC is a diamter then we know that angle in a semi-circle is right angle, which means Angle BAC will be 90 degree. If we draw an altitude from A on side BC, then Area of this triangle will be (1/2)*BC*Altitude. Now its very clear that for area to be maximum, its altitude should also be maximum which is only possible, when the altitude from A meets the center of the circle. And hence using SAS congurance (in Triangle AOB and AOC), we can say that AB = AC – Isosceles triangle.

Thanks.