Question

The hypotenuse of a right triangle is 25 cm. The difference between the lengths of the other two sides of the triangle is 5 cm. The product of the lengths of these sides is __ cm.

Answer 1

$\Rightarrow$ Let one of the side be $x$ and the other be $x+5$

By Pythagoras theorem,

$(25{)}^2=x^2+(x+5{)}^2$

$\Rightarrow$ $625=x^2+x^2+10x+25$

$\Rightarrow$ $625=2x^2+10x+25$

$\Rightarrow$ $2x^2+10x-600=0$

$\Rightarrow$ $x^2+5x-300=0$

$\Rightarrow$ $x^2+20x-15x-300=0$

$\Rightarrow$ $x(x+20)-15(x+20)=0$

$\Rightarrow$ $(x+20)(x-15)=0$

$\Rightarrow$ $x+20=0$ and $x-15=0$

$\therefore$ $x=-20$ and $x=15$

But $x$ cannot be negative.

$\Rightarrow$ Hence, one side $=x=15cm$

$\Rightarrow$ Other side $x+5=15+5=20cm$

$\Rightarrow$ The product of the length of sides $=15cm\times 20cm=300c{m}^2$