Question

# The perimeter of a right triangle is 60 cm. Its hypotenuse is 25 cm. Find the area of the triangle.

Let, ABC be the given right angled triangle such that base $= BC = x{\text{ }}cm$ and hypotenuse $AC = 25cm$ . Perimeter is given to us as $60cm$ . Then,
$AB + BC + AC = 60 \\ \Rightarrow AB + x + 25 = 60 \\ \Rightarrow AB = 35 - x \\$
$A{B^2} + B{C^2} = A{C^2} \\ \Rightarrow {(35 - x)^2} + {x^2} = {25^2} \\ \Rightarrow 1225 + {x^2} - 70x + {x^2} = 625 \\ \Rightarrow 2{x^2} - 70x + 600 = 0 \\ \Rightarrow {x^2} - 35x + 300 = 0 \\ \Rightarrow {x^2} - (15 + 20)x + 300 = 0 \\ \Rightarrow {x^2} - 15x - 20x + 300 = 0 \\ \Rightarrow x(x - 15) - 20(x - 15) = 0 \\ \Rightarrow (x - 20)(x - 15) = 0 \\ \Rightarrow x = 15,20 \\$
If, $x = 20{\text{ then }}AB = 35 - x = 35 - 20 = 15{\text{ and }}BC = x = 20$ .
Area of a triangle can be calculated by $\frac{1}{2}(base \times height)$ so, $\frac{1}{2}(BC \times AB) = \frac{1}{2}(15 \times 20) = 150c{m^2}$ .