What is the largest area, in cm², of a right-angled triangle with hypotenuse 10 cm?

Explanation

1. 01

   With legs $a$ and $b$, $a^{2} + b^{2} = 100$ and the area is $\frac{1}{2}ab = \frac{1}{2}a\sqrt{100 - a^{2}}$.

2. 02

   Its square, $\frac{1}{4}(100a^{2} - a^{4})$, is largest when $a^{2} = 50$: then $b^{2} = 50$ too, and the area is $\frac{1}{2} \times 50 = 25$.