Question

what is the length of \\(\overline{ac}\\)?
\\(\bigcirc\\) 3 ft
\\(\bigcirc\\) 4 ft
\\(\bigcirc\\) 18 ft
\\(\bigcirc\\) 12 ft
(image: triangle abc with m on ab, n on bc, mn = 6 ft, bm = 4 ft, bn = 3 ft, markings indicating midpoints or segment equalities)

Explanation:

Step1: Identify Midline Theorem

In triangle \(ABC\), \(M\) and \(N\) are midpoints (since segments are marked equal), so \(MN\) is the midline. The Midline Theorem states that the midline is parallel to the third side and half its length.

Step2: Apply Midline Theorem

Let \(AC = x\). By Midline Theorem, \(MN=\frac{1}{2}AC\). Given \(MN = 6\) ft, so \(6=\frac{1}{2}x\).

Step3: Solve for \(AC\)

Multiply both sides by 2: \(x = 6\times2=12\) ft.

Answer:

12 ft