Question

what is the length of \\(overline{ac}\\)?
\\(\bigcirc\\) 3 ft
\\(\bigcirc\\) 4 ft
\\(\bigcirc\\) 18 ft
\\(\bigcirc\\) 12 ft

Explanation:

Step1: Identify segment ratio

First, find the ratio of BM to BA:
$BA = BM + MA = 4 + MA$, but we use the ratio of similar triangles:
$\frac{BM}{BA} = \frac{4}{4+MA}$, but since $\triangle BMN \sim \triangle BAC$, $\frac{BM}{BA} = \frac{MN}{AC}$.
First calculate $\frac{BM}{BA} = \frac{4}{4+4} = \frac{1}{2}$? No, wait: total length of BA is $4 + 4 = 8$? No, wait, $\frac{BM}{BA} = \frac{4}{4+4} = \frac{1}{2}$, and $\frac{BN}{BC} = \frac{3}{3+3} = \frac{1}{2}$. So the scale factor of $\triangle BMN$ to $\triangle BAC$ is $\frac{1}{2}$.

Step2: Set up proportion

Since $\frac{MN}{AC} = \frac{1}{2}$, substitute $MN=6$:
$\frac{6}{AC} = \frac{1}{2}$

Step3: Solve for AC

Rearrange to solve for $AC$:
$AC = 6 \times 2 = 12$

Answer:

D. 12 ft