Question

$overline{hi} parallel overline{gj}$. find $fh$.

(image of a triangle with points f, j, i on the base, f, g, h on the left side. fg = 5, fj = 11, ji = 22. hi is parallel to gj.)

Explanation:

Step1: Identify Similar Triangles

Since \( \overline{HI} \parallel \overline{GJ} \), by the Basic Proportionality Theorem (Thales' theorem), \( \triangle F G J \sim \triangle F H I \). So the ratios of corresponding sides are equal: \( \frac{FG}{FH}=\frac{FJ}{FI} \).

Step2: Calculate \( FI \)

\( FI = FJ + JI = 11 + 22 = 33 \). Let \( FH = x \), then \( FG = x - 5 \) (since \( FG = FH - GH \) and \( GH = 5 \)).

Step3: Set Up Proportion

Substitute into the proportion: \( \frac{x - 5}{x}=\frac{11}{33} \). Simplify \( \frac{11}{33}=\frac{1}{3} \), so \( \frac{x - 5}{x}=\frac{1}{3} \).

Step4: Solve for \( x \)

Cross - multiply: \( 3(x - 5)=x \). Expand: \( 3x-15 = x \). Subtract \( x \) from both sides: \( 2x-15 = 0 \). Add 15 to both sides: \( 2x = 15 \)? Wait, no, wait. Wait, \( 3x-15=x \), subtract \( x \): \( 2x = 15 \) is wrong. Wait, \( 3x-15=x \), then \( 3x - x=15 \), \( 2x = 15 \)? No, wait, let's re - do. Wait, \( \frac{x - 5}{x}=\frac{11}{33}=\frac{1}{3} \). Cross - multiply: \( 3(x - 5)=x \), \( 3x-15=x \), \( 3x - x=15 \), \( 2x = 15 \)? No, that can't be. Wait, maybe I made a mistake in \( FG \). Wait, \( FG = 5 \), \( FH=x \), so \( \frac{FG}{FH}=\frac{5}{x} \)? Wait, no! Wait, the triangles: \( \triangle FGJ \) and \( \triangle FHI \). So the corresponding sides: \( FG \) corresponds to \( FH \), \( FJ \) corresponds to \( FI \). So \( FG = 5 \), \( FH=x \), \( FJ = 11 \), \( FI=11 + 22=33 \). So the correct proportion is \( \frac{FG}{FH}=\frac{FJ}{FI} \), so \( \frac{5}{x}=\frac{11}{33} \). Ah! There was my mistake. I had \( FG=x - 5 \), but actually \( FG = 5 \), \( FH=x \), so \( FG = 5 \), \( FH=x \). So the proportion is \( \frac{5}{x}=\frac{11}{33} \).

Step5: Correct Proportion and Solve

Simplify \( \frac{11}{33}=\frac{1}{3} \), so \( \frac{5}{x}=\frac{1}{3} \). Cross - multiply: \( x = 5\times3=15 \). Wait, but let's check. If \( FH = 15 \), then \( FG = 5 \), \( FJ = 11 \), \( FI = 33 \). \( \frac{5}{15}=\frac{1}{3} \), \( \frac{11}{33}=\frac{1}{3} \). Yes, that works. So my initial mistake was in identifying \( FG \). So the correct proportion is \( \frac{FG}{FH}=\frac{FJ}{FI} \) with \( FG = 5 \), \( FH=x \), \( FJ = 11 \), \( FI = 33 \).

Answer:

\( 15 \)