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

I know the slope and length of $c$, how do I find the length of $a$ and $b$?
 A: 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 $a = bm$.

A: 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
$$b = \frac{c}{\sqrt{m^2 + 1}}$$
A: if you kno9w the overall slope, but maybe you want to find the Y value of a shorter distance for x, use these relations.
where F is the hypotenuse, X is the x value, and Y is the Y value. 
Fcos(angle)=X,
Fsin(angle)=Y,
which you probably have memorized, but here are the ones you've forgotten. 
Xtan(angle)=Y,
Ycot(angle)=X,  
A: $m = \frac{a}{b}$
If you take your simplified $a$ and $b$, and use $a^2+b^2=c^2$, you can do $\frac{hypotenuse}{c}$ to get the scaling factor. Once you have your scaling factor, put your slope into fraction form.
Multiply your numerator by the scaling factor and you will get $a$. Multiply your denominator by the scaling factor and you will get $b$.
