Question

what is the length of \overline{dc}?
2 units
3 units
6 units
9 units

Explanation:

Step1: Use the mid - segment theorem

The mid - segment theorem states that if a segment (here \(DE\)) is a mid - segment of a triangle (\(\triangle ABC\), since \(DE\parallel BC\) and \(DE\) connects the mid - points of two sides of the triangle), then \(DE=\frac{1}{2}BC\) and \(\frac{AD}{AC}=\frac{AE}{AB}\). Also, \(\frac{AD}{DC}=\frac{AE}{EB}\).

We know that \(AE = 12\), \(EB=4\), and \(AD = 9\). Let \(DC=x\).

Step2: Set up the proportion

By the basic proportionality theorem (Thales' theorem), \(\frac{AD}{DC}=\frac{AE}{EB}\). Substitute the given values: \(\frac{9}{x}=\frac{12}{4}\).

Step3: Cross - multiply

Cross - multiplying gives us \(12x=9\times4\).

Step4: Solve for \(x\)

\(12x = 36\), then \(x=\frac{36}{12}=3\).

Answer:

3 units