What is the area of a right-angled triangle whose hypotenuse is 50 cm?

It is known that the lengths of sides of a right-angled triangle form a Pythagorean triplet.

The numbers 2m, m² − 1, and m² + 1, where m > 1, form a Pythagorean triplet.

$$\begin{gathered} m^2+1=50 \\ \Rightarrow m^2=49 \\ \Rightarrow m=7 \\ \Rightarrow 2m=14 \end{gathered}$$

m² − 1 = 7² − 1 = 49 − 1 = 48

Therefore, the other perpendicular sides of the triangle are 14 cm and 48 cm.

Area of a triangle is given by: $\frac{1}{2}\times \mathrm{Base}\times \mathrm{Height}$

$\begin{array}{rcl}\mathrm{Area} \mathrm{of} \mathrm{triangle}& =& \frac{1}{2}\times 14\times 48\\ & =& 7\times 48\\ & =& 336 {\mathrm{cm}}^2\end{array}$

Thus, the area of the given triangle is 336 cm².