Question

in the diagram below, $\angle fde \cong \angle fgh$. $gd = 1.5$, $he = 2.6$, and $fg = 2.5$. find the length of $\overline{fh}$. round your answer to the nearest tenth if necessary.

Explanation:

Step1: Identify Similar Triangles

Since \( \angle FDE \cong \angle FGH \) and \( \angle F \) is common to both \( \triangle FDE \) and \( \triangle FGH \), by the AA (Angle - Angle) similarity criterion, \( \triangle FGH \sim \triangle FDE \). This means the corresponding sides of the similar triangles are proportional. So, we have the proportion \( \frac{FG}{FD}=\frac{FH}{FE} \).

First, we need to find the lengths of \( FD \) and \( FE \) in terms of \( FG \), \( GD \), \( FH \), and \( HE \). We know that \( FD = FG+GD \) and \( FE=FH + HE \). Given that \( FG = 2.5 \), \( GD = 1.5 \), and \( HE=2.6 \), we can substitute these values into the proportion. So \( FD=2.5 + 1.5=4 \) and \( FE=FH + 2.6 \).

The proportion becomes \( \frac{2.5}{4}=\frac{FH}{FH + 2.6} \).

Step2: Solve the Proportion for \( FH \)

Cross - multiply the proportion \( \frac{2.5}{4}=\frac{FH}{FH + 2.6} \). Cross - multiplying gives us \( 2.5\times(FH + 2.6)=4\times FH \).

Expand the left - hand side: \( 2.5FH+2.5\times2.6 = 4FH \). Calculate \( 2.5\times2.6 = 6.5 \), so the equation is \( 2.5FH+6.5 = 4FH \).

Subtract \( 2.5FH \) from both sides of the equation: \( 6.5=4FH - 2.5FH \).

Simplify the right - hand side: \( 4FH-2.5FH = 1.5FH \). So we have the equation \( 1.5FH=6.5 \).

Now, solve for \( FH \) by dividing both sides of the equation by \( 1.5 \): \( FH=\frac{6.5}{1.5}=\frac{13}{3}\approx4.3 \) (rounded to the nearest tenth).

Answer:

\( 4.3 \)