Triangles $DEF$ and $GHF$ are similar triangles. $GD = 16$ in. $DE = 20$ in. $FH = 20$ in. We need to find the length of $EF$.

1. Problem Description

Triangles $DEF$ and $GHF$ are similar triangles.

$GD = 16$ in.

$DE = 20$ in.

$FH = 20$ in.

We need to find the length of $EF$.

2. Solution Steps

Since triangles $DEF$ and $GHF$ are similar, their corresponding sides are proportional. Therefore,

$\frac{HF}{EF} = \frac{GF}{DF} = \frac{GH}{DE}$

Also, $DF = DG + GF$, so $GF = DF - DG$.

From $\frac{HF}{EF} = \frac{DE}{EF - FH} = \frac{DF}{DE}$, we know that triangles $GHF$ and $DEF$ share the angle at $F$. We also know that the triangles are similar.

Therefore, we have the following ratios:

$\frac{HF}{EF} = \frac{GF}{DF}$

$\frac{HF}{EF} = \frac{GH}{DE}$

$\frac{GF}{DF} = \frac{GH}{DE}$

Given:

$DE = 20$ in.

$HF = 20$ in.

$DG = 16$ in.

Let $x = EF$.

We need to find the length of $EF$.

Since $GHF$ and $DEF$ are similar triangles,

$\frac{HF}{EF} = \frac{GH}{DE}$

$\frac{FH}{EF} = \frac{FG}{FD} = \frac{HG}{ED}$

We are given $FH = 20$, $ED = 20$. Let $EF = x$.

From the diagram, we have:

$\frac{HF}{EF} = \frac{20}{x}$

$\frac{GH}{DE} = \frac{DG}{DE} = \frac{16}{20}$

Let's use $\frac{HF}{EF} = \frac{GH}{DE}$, but this doesn't work because we do not know HG.

Using similar triangles, we can write the ratio $\frac{DE}{GH} = \frac{EF}{FH} = \frac{DF}{GF}$.

Given that $DE = 20$, $FH = 20$, $DG = 16$. We want to find $EF$.

Also, $DE/DG = 20/16 = 5/4$.

$\frac{DE}{DG+GE} = \frac{EF}{FH}$

$\frac{DG}{DE} = \frac{16}{20} = \frac{4}{5}$

Therefore, $\frac{FH}{EF} = \frac{4}{5}$.

$EF = \frac{5}{4} FH = \frac{5}{4} (20) = 25$

3. Final Answer

25