The altitude drawn to the hypotenuse of a right triangle divides the hypotenuse into segments such as their lengths are in ratio of 1: 4. if the length of the altitude is 8, find the length of: (a) each segment of the hypotenuse (b) the longer leg of the triangle.

I happen to know that when the foot F of the altitude divides the hypotenuse into two lengths and , the altitude is the geometric mean of and , that is

But you probably didn't know that. We'll take our triangle labeled the usual way, right angle C, hypotenuse . There are three right triangles, which I'll write without the benefit of the figure. Let's say sides and so

Adding all three equations lots of stuff cancels,

That proves the geometric mean thing. In our problem we have which just means .

Part a)

Part b)

Check:

The triangle is so

a 90 degree rotation across the the origin followed by a reflection across the x-axis.

step-by-step explanation: