Question

which value of x would make (overline{fg}paralleloverline{bc})?
1
3
6
9

Explanation:

Step1: Apply the basic proportionality theorem (Thales' theorem)

If \(FG\parallel BC\), then \(\frac{AF}{FB}=\frac{AG}{GC}\).
Substitute \(AF = x + 1\), \(FB=x + 6\), \(AG=x - 1\), and \(GC=x + 3\) into the proportion: \(\frac{x + 1}{x + 6}=\frac{x - 1}{x + 3}\).

Step2: Cross - multiply

\((x + 1)(x + 3)=(x - 1)(x + 6)\).
Expand both sides:

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Step3: Simplify the equation

Subtract \(x^{2}\) from both sides: \(4x + 3=5x-6\).
Subtract \(4x\) from both sides: \(3=x - 6\).
Add \(6\) to both sides: \(x=9\).

Answer:

\(9\)