## Precalculus: Mathematics for Calculus, 7th Edition

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.