# Thread: function of the lenght of the hypotenuse

1. ## function of the length of the hypotenuse

One of the legs of a right triangle has length 4cm. Express the length of the altitude perpendicular to the hypotenuse as a function of the length of the hypotenuse.

I know this divides the Right Triangle into 2 other Right Triangles with an shared side but couldn't see how to set up the equation using this, that is if that is the best way to do it.

The answer to this is $a=\frac{4\sqrt{h^2-16}}{h}$
where $a$ is the length of the altitude and $h$ is the length of the hypotenuse

2. Right... you need to see that there are two congruent triangles. They share a common angle at the top vertex and they have both a right angle. So, they are congruent.

So, next, you take the two longest sides of the triangles, the hypotenuses. The shorter hypotenuse over the longer hypotenuse should be equal to a shorter side over a longer side of two corresponding sides of both triangles.

For this, I will take the side a. This will correspond to the last dies of the larger triangle. And this side is given by Pythagoras' Theorem. Let's call this side x.

$\frac{4}{h} = \frac{a}{x}$

x is given by h^2 = 4^2 + x^2, so x = $\sqrt{h^2 - 4^2}$

Substitute that in your above equation and you are done after some manipulation