Right triangle trig question. respond with a legitimate answer and with a thorough explanation.

I can think of three ways. I'll try them all.

This is the construction for the geometric mean of and , so .  That's no fun, but I suppose it's nice to know the answer going in.

We have three similar triangles, because the two little ones have a common angle and a right angle, so same angles as the big one.





So









There it is again.

I guess the way they really want you to do it is to write the Pythagorean Theorem a few times.  Let's abbreviate







is our unknown.  Adding all three equations,







That's three different ways.  Plugging in the numbers,

fourth choice.

I hate it when they ruin a nice exact answer with an approximation.