Answer
Please see step-by-step.
Work Step by Step
Being in arithmetic progression, the sides have a common difference d, so we label
the lengths of the three sides,\ from shortest to longest, with
$x-d,\ x$, and $x+d$
The triangle is a right triangle,
$x+a$ is the longest side, so it is the hypotenuse.
Apply the Pythagorean Theorem:
$(x-d)^{2}+x^{2}=(x+d)^{2}$
$x^{2}-2dx+d^{2}+x^{2}=x^{2}+2dx+d^{2}$
$x^{2}-4dx=0$
$x(x-4d)=0$
... discard x=0, as it can not be the side of a triangle,
$x=4d$
So, the three sides have lengths
$x-d=4d-d=3d$,
$x=4d$, and
$x+d=4d+d=5a$,
which are proportional to sides 3, 4, and 5,
so the triangle is similar to a 3-4-5 triangle.
