Question

in $\triangle feg$, point $h$ is between points $e$ and $f$, point $j$ is between points $f$ and $g$, and $\overline{hj} \parallel \overline{eg}$.
$eh = 8$, $hf = 12$, and $fg = 30$.
what is $fj$?
enter your answer in the box.
$fj = \square$

Explanation:

Step1: Apply the Basic Proportionality Theorem (Thales' theorem)

Since \( \overline{HJ} \parallel \overline{EG} \) in \( \triangle FEG \), by the Basic Proportionality Theorem, we have \( \frac{EH}{HF}=\frac{EJ}{JG} \), but actually, the correct proportion for the segments on the sides is \( \frac{EH}{EF}=\frac{FJ}{FG} \)? Wait, no, let's correct that. The correct proportion is \( \frac{HF}{EF}=\frac{FJ}{FG} \) because \( H \) is on \( EF \) and \( J \) is on \( FG \), and \( HJ \parallel EG \). First, find \( EF \): \( EF = EH + HF = 8 + 12 = 20 \).

Step2: Set up the proportion

So, \( \frac{HF}{EF}=\frac{FJ}{FG} \). Substitute the known values: \( HF = 12 \), \( EF = 20 \), \( FG = 30 \). So the equation is \( \frac{12}{20}=\frac{FJ}{30} \).

Step3: Solve for \( FJ \)

Cross - multiply: \( 20\times FJ=12\times30 \). Then \( 20FJ = 360 \). Divide both sides by 20: \( FJ=\frac{360}{20}=18 \).

Answer:

\( FJ = 18 \)