Find the area of a right-angled triangle, if the radius of its circumcircle is 2.5 cm and the altitude
drawn to the hypotenuse is 2 cm long.

Dear Student ,

Please find below required solution

Let ABC be a right angled triangle inside a circle having center 'O'

Given, OA = OB = OC = 2.5 cm  [OA, OB and OC be the radius of circle]

Let AD be the altitude drawn from the opposite vertex to the hypotenuse 

∴ AD = 2 cm

Since, ΔABC is a right triangle right angled at A 

∴ A circle can be drawn which passes through the vertices of ΔABC  (Angle in a semi-circle is 90°)

 $$\begin{gathered} ar(∆ABC) = \frac{1}{2} x BC\times AD \\ ar(∆ABC) = \frac{1}{2} x (OB +OC )\hspace{0.17em} \times AD \\ ar(∆ABC) = \frac{1}{2} x (2.5+2.5 )\hspace{0.17em} \times 2 = 5 c{m}^2 \end{gathered}$$

Regards