I need help on right triangles

Question

With the premise that the length of a right triangle's hypotenuse is 5.822153, what is the length of its two other sides such that if the triangle's hypotenuse were its base, then the height of the triangle will be 1.987424?

Answer 1

It can easily be proved using Similarity of Triangles that $h^2 = x(c-x)$.

You have the value of $h$ and $c$. Solve the resulting quadratic to get $x$ and apply Pythagoras Theorem to get the two sides of the triangle.

Answer 2

Based on @hardmath's comment, if you let the catheti (perpendicular sides) have lengths $a$ and $b$ and the hypotenuse $c$ and height (perpendicular to hypotenuse) $h$, you get the system of two equations:

$\frac 12 ab = \frac 12 ch$ (based on equal areas computed two different ways)

$a^2 + b^2 = c^2$ (based on Pythagoras' theorem)

Putting $b^2 = c^2 - a^2$ and squaring the first equation, you get:

$a^2(c^2 - a^2) = c^2h^2$

Rearranging, you get a quartic in $a$ (which is a quadratic in $a^2$):

$(a^2)^2 - c^2\cdot a^2 + c^2h^2 = 0$

and you can solve that for $a^2$, from which you can compute $a$.