Question

if h is between g and f, and ( gh = 3x ), ( hf = 6 ), and ( gf = 18 ), find the value of x and the length of ( overline{gh} ).

Explanation:

Step1: Use the segment - addition postulate

Since \(H\) is between \(G\) and \(F\), by the segment - addition postulate \(GF = GH+HF\).
We know that \(GF = 18\), \(GH = 3x\), and \(HF = 6\). So the equation is \(3x+6=18\).

Step2: Solve the equation for \(x\)

Subtract \(6\) from both sides of the equation \(3x+6 = 18\).
We get \(3x+6 - 6=18 - 6\), which simplifies to \(3x=12\).
Then divide both sides by \(3\): \(\frac{3x}{3}=\frac{12}{3}\), so \(x = 4\).

Step3: Find the length of \(GH\)

Substitute \(x = 4\) into the expression for \(GH\). Since \(GH = 3x\), then \(GH=3\times4 = 12\).

Answer:

1. \(x = 4\)
2. \(GH=12\)