Question

Question was provided via image upload.

Explanation:

Step1: Identify the property

In a triangle, if a segment from a vertex is perpendicular to the opposite side and the two adjacent sides are congruent (marked with ticks), then the perpendicular segment bisects the opposite side. So \( EG = FG \).

Step2: Set up the equation

Since \( EG = 2x + 11 \) and \( FG = 14x - 37 \), we set them equal: \( 2x + 11 = 14x - 37 \).

Step3: Solve for x

Subtract \( 2x \) from both sides: \( 11 = 12x - 37 \).
Add 37 to both sides: \( 48 = 12x \).
Divide by 12: \( x = 4 \).

Step4: Find EG

Substitute \( x = 4 \) into \( EG = 2x + 11 \): \( EG = 2(4) + 11 = 8 + 11 = 19 \).

Answer:

\( x = 4 \), \( EG = 19 \)