How to find the side lengths of a triangle given the hypotenuse length and tangent?

Question

Let's say there is a triangle with hypotenuse length 225 and tangent ratio 15/8, is there a method short of trial and error to find the length of the opposite and adjacent sides?

Answer 1

Your sides are $15x$ and $8x$.

Using the Pythagorean Theorem you have $ (15x)^2 + ( 8x)^2 =225^2.$

That results in $x^2 = \frac {225^2}{289} $ or $x=13.235$

Thus the sides are approximately $105.88$ and $198.52$

Answer 2

$a^2 + b^2 = 225\\ \frac {b}{a} = \frac {18}{5}\\ b = \frac {18}{5}a\\ b^2 = \frac {18^2}{5^2}a^2\\ a^2+b^2 = \frac {18^2+5^2}{5^2} a^2 = 225\\ 225 = 15^2\\ a^2 = \frac {15^2\cdot 5^2}{18^2+5^2}\\ a = \frac {5\cdot 15}{\sqrt {18^2+5^2}}\\ b = \frac {18\cdot 15}{\sqrt {18^2+5^2}}$

Answer 3

We can draw 2 triangles with big side parallel to hypotenuse $225$ sides $8,17$

$$=\sqrt {15^2+8^2} = 17 $$

Magnification ratio $$ m=\dfrac{225}{17}$$

So the sides are

$$ 8m, 15m \, .$$