Question

find gj.
gj = \boxed{}

Explanation:

Step1: Identify Similar Triangles

The lines \( HG \) and \( IK \) have arrows, indicating they are parallel. So, \( \triangle HJG \sim \triangle IJK \) by the Basic Proportionality Theorem (Thales' theorem).

Step2: Set Up Proportion

For similar triangles, the ratios of corresponding sides are equal. So, \( \frac{HJ}{IJ} = \frac{GJ}{KJ} \). We know \( HJ = 30 \), \( IJ = 15 \), and \( KJ = GJ - 14 \) (since \( GK = 14 \)). Let \( GJ = x \), then \( KJ = x - 14 \). Substituting values: \( \frac{30}{15} = \frac{x}{x - 14} \).

Step3: Solve the Proportion

Simplify \( \frac{30}{15} = 2 \), so \( 2 = \frac{x}{x - 14} \). Cross - multiply: \( 2(x - 14)=x \). Expand: \( 2x - 28 = x \). Subtract \( x \) from both sides: \( 2x - x - 28 = 0 \), so \( x = 28 \).

Answer:

\( 28 \)