which value of x would make (overline{fg}paralleloverline{bc})?\n1\n3\n6\n9

Explanation:

Step1: Apply the basic - proportionality theorem

If $\overline{FG}\parallel\overline{BC}$, then $\frac{AF}{FB}=\frac{AG}{GC}$. Here, $AF = x + 1$, $FB=x + 6$, $AG=x - 1$, and $GC=x + 3$. So, $\frac{x + 1}{x + 6}=\frac{x - 1}{x + 3}$.

Step2: Cross - multiply

Cross - multiplying gives $(x + 1)(x + 3)=(x - 1)(x + 6)$. Expanding both sides: [ \begin{align*} x^{2}+3x+x + 3&=x^{2}+6x-x - 6\ x^{2}+4x + 3&=x^{2}+5x - 6 \end{align*} ]

Step3: Simplify the equation

Subtract $x^{2}$ from both sides of the equation: $4x + 3=5x - 6$. Then, move the $x$ terms to one side and constants to the other side: $5x-4x=3 + 6$.

Step4: Solve for x

$x=9$.

Answer:

$9$