Question

1. given: (overline{pq}paralleloverline{bc}), find the length of (overline{aq}).

Explanation:

Step1: Use the basic - proportionality theorem

Since $\overline{PQ}\parallel\overline{BC}$, we have $\frac{AP}{PB}=\frac{AQ}{QC}$. Given $AP = 6$, $PB = 12$, and $QC = 18$. Let $AQ=x$. Then $\frac{6}{12}=\frac{x}{18}$.

Step2: Cross - multiply to solve for $x$

Cross - multiplying the equation $\frac{6}{12}=\frac{x}{18}$ gives us $12x=6\times18$.

Step3: Simplify the right - hand side

$6\times18 = 108$, so the equation becomes $12x = 108$.

Step4: Solve for $x$

Dividing both sides of the equation $12x = 108$ by 12, we get $x=\frac{108}{12}=9$.

Answer:

9