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 $EG=\sqrt{ab}$.  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.

$\dfrac{a}{DE} = \dfrac{EG}{EF} = \dfrac{DE}{a+b}$

$\dfrac{EG}{DE} = \dfrac{b}{EF} = \dfrac{EF}{a+b}$

So

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

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

$EG = \dfrac{b \ DE}{EF}$

$EG^2 = \dfrac{b^2 DE^2}{EF^2} = \dfrac{ab^2(a+b)}{b(a+b)}=ab$

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,

$\sqrt{2(16)}=4\sqrt{2} \approx 5.657 \quad$ fourth choice.

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