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
