Question

1. what is the correct equation for segment ef?

options:
$overline{ef} = sqrt{(eg)(eh)}$
$overline{ef} = sqrt{(eg^2)(fg^2)}$
$overline{ef} = sqrt{(eg)(gh)}$
$overline{ef} = sqrt{(eg^2)(gh^2)}$

Explanation:

Step1: Identify similar triangles

In right triangle $EFH$ with altitude $FG$ to hypotenuse $EH$, $\triangle EFG \sim \triangle EHF$ by AA similarity (shared $\angle E$, right angles).

Step2: Set up proportion from similarity

From similar triangles: $\frac{EF}{EH} = \frac{EG}{EF}$

Step3: Cross-multiply to solve for $EF$

Cross-multiplying gives $EF^2 = EG \times EH$, so $EF = \sqrt{(EG)(EH)}$.
First calculate $EH$: $EH = EG + GH = 6 + 7 = 13$ m, but the equation only needs the segment terms.

Answer:

$\overline{EF} = \sqrt{(EG)(EH)}$