CAT 2023 Slot 2 QA Question 18: A triangle is drawn with its vertices on the circle $C$ such that one of its sides is a diameter of $C$ and the other two sides have their lengths in the ratio of $a : b$. If the radius of the circle

Question

A triangle is drawn with its vertices on the circle $C$ such that one of its sides is a diameter of $C$ and the other two sides have their lengths in the ratio of $a : b$. If the radius of the circle

Accepted Answer

A triangle whose one side is a diameter of its circumcircle is right-angled at the opposite vertex.

Let the two non-diameter sides be $ax$ and $bx$. The diameter is $2r$.
$(ax)^{2} + (bx)^{2} = (2r)^{2}$
$x^{2}(a^{2} + b^{2}) = 4r^{2}$
$x^2 = \dfrac{4r^2}{a^2 + b^2}$

The area of the right triangle

A triangle is drawn with its vertices on the circle CCC such that one of its sides is a diameter of CCC and the other two sides have their lengths in the ratio of a:ba : ba:b. If the radius of the circle is rrr, then the area of the triangle is

A triangle is drawn with its vertices on the circle $C$ such that one of its sides is a diameter of $C$ and the other two sides have their lengths in the ratio of $a : b$ . If the radius of the circle is $r$ , then the area of the triangle is