Question

find hj.
hj =
submit

Explanation:

Step1: Identify Similar Triangles

From the diagram, \( FJ \parallel GJ \) (marked with tick marks) and \( FI \parallel GH \) (marked with arrows), so \( \triangle FJI \sim \triangle GJH \) by the Basic Proportionality Theorem (Thales' theorem) or AA similarity (corresponding angles equal).

Step2: Set Up Proportion

The ratio of corresponding sides in similar triangles is equal. So, \( \frac{FJ}{GJ} = \frac{JI}{JH} \). We know \( FJ = 42 \), \( GJ = 63 \), and \( IH = 23 \). Let \( JI = x \), then \( JH = x + 23 \). The ratio \( \frac{42}{63}=\frac{2}{3} \). So, \( \frac{x}{x + 23}=\frac{2}{3} \). Cross - multiply: \( 3x=2(x + 23) \).

Step3: Solve for \( x \)

Expand the right - hand side: \( 3x = 2x+46 \). Subtract \( 2x \) from both sides: \( 3x-2x=2x + 46-2x \), so \( x = 46 \).

Step4: Find \( HJ \)

Since \( HJ=JI + IH \) and \( JI = 46 \), \( IH = 23 \), then \( HJ=46 + 23=69 \).

Answer:

\( 69 \)