Right Angle Triangle Find hypotenuse?

Question

In a right angel triangle, one of its perpendicular side is $10$cm less than the other and the shortest side is $15$cm less than the hypotenuse? Find the length of the hypotenuse.

I have done it like this: one of perpendicular side $x+10$, other perpendicular side $x$, then hypotenuse $x+15$, but by using this I'm not getting the right answer??

and in book they have done like this one of perpendicular side $x-5$, other perpendicular side $x-15$, then hypotenuse $x$.

Can u Plz tell me What is wrong with my approach

Answer 1

There is nothing wrong with your approach. Let $x$ be the shortest side. Then, the other perpendicular side has length $x+10$ and the hypotenuse has length $x+15$. Thus, by the Pythagorean theorem,

$$ \begin{aligned} x^{2}+(x+10)^{2}=(x+15)^{2}&\implies x^{2}+x^{2}+20x+100=x^{2}+30x+225\\ &\implies x^{2}-10x-125=0\\ &\implies x=\frac{10\pm\sqrt{100+500}}{2}=5\pm 5\sqrt{6} \end{aligned} $$

Since $x$ must be positive, $x=5+5\sqrt{6}$. Therefore, the length of the hypotenuse is $20+5\sqrt{6}$.