Question

if \\(\overline{gj} \cong \overline{ij}\\), \\(gh = 2t + 77\\), and \\(hi = 9t\\), what is \\(gh\\)?
\\(gh = \square\\)

Explanation:

Step1: Use the perpendicular bisector theorem

Since \( \overline{GJ}\cong\overline{IJ}\) and \( \overline{HJ}\perp\overline{GI}\), by the perpendicular bisector theorem, \(GH = HI\).

Step2: Set up the equation

Set \(2t + 77=9t\).

Step3: Solve for \(t\)

Subtract \(2t\) from both sides: \(77 = 9t-2t\), so \(77 = 7t\). Then \(t=\frac{77}{7}=11\).

Step4: Find \(GH\)

Substitute \(t = 11\) into \(GH=2t + 77\). So \(GH=2\times11+77=22 + 77=99\).

Answer:

\(99\)