Question

find fh.
69
f 23 j i
g
34
h
fh =
submit

Explanation:

Step1: Identify Similar Triangles

The segments \( GJ \) and \( HI \) are parallel (indicated by the arrows), so \( \triangle FGJ \sim \triangle FHI \) by the Basic Proportionality Theorem (Thales' theorem). This means the ratios of corresponding sides are equal: \( \frac{FJ}{FI} = \frac{FG}{FH} \).

Step2: Find the Ratio of Sides

First, find the ratio of \( FJ \) to \( FI \). Given \( FJ = 23 \) and \( FI = 69 \), the ratio is \( \frac{23}{69} = \frac{1}{3} \).

Step3: Set Up the Proportion for \( FH \)

Let \( FH = x \). We know \( FG = x - 34 \) (since \( GH = 34 \)). From the similarity ratio: \( \frac{FG}{FH} = \frac{1}{3} \), so \( \frac{x - 34}{x} = \frac{1}{3} \).

Step4: Solve for \( x \)

Cross - multiply: \( 3(x - 34)=x \). Expand: \( 3x-102 = x \). Subtract \( x \) from both sides: \( 2x - 102=0 \). Add 102 to both sides: \( 2x = 102 \). Divide by 2: \( x = 51 \).

Answer:

\( 51 \)