Question

point q is the midpoint of \\( \overline { g h } \\). \\( g q = 2 x + 3 \\), and \\( g h = 5 x - 5 \\) what is the length of \\( \overline { g q } \\)? enter your answer in the box units

Explanation:

Step1: Use the midpoint property

Since \(Q\) is the midpoint of \(\overline{GH}\), then \(GH = 2GQ\).
Given \(GQ=2x + 3\) and \(GH = 5x-5\), we substitute into the equation \(5x-5=2(2x + 3)\).

Step2: Solve the equation for \(x\)

Expand the right - hand side: \(5x-5 = 4x+6\).
Subtract \(4x\) from both sides: \(5x-4x-5=4x - 4x+6\), which gives \(x-5=6\).
Add \(5\) to both sides: \(x=6 + 5=11\).

Step3: Find the length of \(GQ\)

Substitute \(x = 11\) into the formula for \(GQ\): \(GQ=2x+3\).
\(GQ=2\times11+3=22 + 3=25\).

Answer:

\(25\)