Answer
$5~\text{in},~12~\text{in},~13~\text{in}~$
Work Step by Step
Write the Pythagorean theorem and solve for $x$:
$$x^2+(x-7)^2=(x+1)^2$$
$$x^2+x^2-14x+49=x^2+2x+1$$
$$x^2-16x+48=0$$
Using Vieta's theorem sum of the roots is $16$ and product is $48$.
Such numbers are $12$ and $4$. So, solutions are:
$x_1=12$ and $x_2=4$
But note, that $4$ cannot be solution as $x-7$ becomes negative and length cannot have negative value.
So, answer is $12$
$$x=12$$
$$x-7=12-7=5$$
$$x+1=12+1=13$$
