In a right triangle, given slope and length of hypotenuse find length of legs.

We have a right triangle, so there are two things we know:

• Slope $\;m = \dfrac{a - 0}{b-0}=\dfrac ab\implies a = bm$.

And

• $a^2 + b^2 = \underbrace{c^2}_{\text{hypotenuse}}$

Two equations and two unknowns.

SPOILER ALERT:

Since $a = bm, $ we can substitute $bm$ into the variable $a$ in the second equation: $$(bm)^2 + b^2 = c^2\implies b^2(m^2 + 1) = c^2 \implies b^2 = \dfrac{c^2}{m^2 + 1} \implies b = \dfrac{c}{\sqrt{m^2 + 1}}.$$ Since the lengths of the sides of a triangle must be positive, we can take the positive root of $b^2$ to solve for $b$, then back substitute to obtain $a = bm$.

If you have the "slope" $$m = \frac ab$$ then you can write $a$ as $mb$. Fit this in $$c = \sqrt{a^2 + b^2}$$ and get $$b = \frac{c}{\sqrt{m^2 + 1}}$$