I'm looking for a formula (or set of formulas) that would be able to determine the $A$, or $B$ value given a right triangle when only $C$ and the ratio $A:B$ is known. I want this mostly for personal use with PC monitors and other boxy shape calculations, but I may find use for it in a project in the future. Anyways, lets use an example
Say I have a right triangle: we know that $C$ (the hypotenuse) is $23.6$ inches. We also know that $A:B=16:9$. How can we solve for $A$ or $B$?
Taken from comments:
$A^2 + B^2 =C^2$ and $\dfrac AB=k$, where $C$ and $k$ are known. Then $A=kB$. So $(kB)^2+B^2=C^2\implies B^2(1+k^2)=C^2\implies B=\dfrac{C}{\sqrt{1+k^2}}$ and $A=\dfrac{kC}{\sqrt{1+k^2}}$.