Question

Right Triangle Trig question. Please respond with a legitimate answer and with a thorough explanation.

Answer 1

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

This is the construction for the geometric mean of
`a` and
`b`, so
`EG=√(ab)`. 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.


`(a)/(DE) = (EG)/(EF) = (DE)/(a+b)`


`(EG)/(DE) = (b)/(EF) = (EF)/(a+b)`

So


`EF^2 = b(a+b)`


`DE^2 = a(a+b)`


`EG = (b \ DE)/(EF)`


`EG^2 = (b^2 DE^2)/(EF^2) = (ab^2(a+b))/(b(a+b))=ab`

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
`d=ED, f=EF, g=EG`


`g^2 + a^2 = d^2`


`g^2 + b^2 = f^2`


`d^2 + f^2 = (a+b)^2`


`g` is our unknown. Adding all three equations,


`g^2 + a^2 + g^2 + b^2+ d^2 + f^2= d^2 + f^2 + a^2 + b^2 + 2ab`


`2g^2=2ab`


`g^2=ab`

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


`√(2(16))=4√(2) \approx 5.657 \quad` fourth choice.

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