Question

point h is on line segment (overline{gi}). given (gh = x + 6), (hi = x + 5), and (gi = 3x + 8), determine the numerical length of (overline{gh}).

Explanation:

Step1: Use segment addition postulate

Since \( H \) is on \( \overline{GI} \), we have \( GH + HI = GI \). Substituting the given expressions: \( (x + 6) + (x + 5) = 3x + 8 \).

Step2: Simplify and solve for \( x \)

Simplify left side: \( 2x + 11 = 3x + 8 \). Subtract \( 2x \) and \( 8 \) from both sides: \( 11 - 8 = 3x - 2x \), so \( x = 3 \).

Step3: Find length of \( \overline{GH} \)

Substitute \( x = 3 \) into \( GH = x + 6 \): \( GH = 3 + 6 = 9 \).

Answer:

\( 9 \)