Question

what is the length of \\( \overline { b c } \\)?
from the markings on the diagram, we can tell \\( e \\) is the midpoint of \\( \overline { b c } \\) and is the midpoint of \\( \overline { a c } \\)
we can apply the theorem: \\( e d = \frac { 1 } { 2 } b a \\).
substituting in the expressions for the lengths and solving for \\( x \\), we get \\( x = \\).
now, since \\( b e = x \\), then \\( b c = \\)

Explanation:

Step1: Identify the mid - segment

From the markings, \(D\) is the midpoint of \(\overline{AC}\). We apply the mid - segment theorem.

Step2: Set up the equation

According to the mid - segment theorem \(ED=\frac{1}{2}BA\). Substitute \(ED = x + 2\) and \(BA=4x - 6\) into the equation:

$$x + 2=\frac{1}{2}(4x - 6)$$

Step3: Solve the equation

Multiply both sides by \(2\) to get \(2(x + 2)=4x-6\).
Expand: \(2x+4 = 4x - 6\).
Subtract \(2x\) from both sides: \(4=2x - 6\).
Add \(6\) to both sides: \(10 = 2x\).
Divide by \(2\): \(x = 5\).

Step4: Find the length of \(BC\)

Since \(E\) is the midpoint of \(BC\) and \(BE=x\), then \(BC = 2BE\). Substitute \(x = 5\) into \(BC\), so \(BC=2\times5=10\).

Answer:

\(x = 5\), \(BC = 10\)