Question

ling an unknown length
what is the length of \\(\overline{bc}\\)?
from the markings on the diagram, we can tell e is the midpoint of \\(\overline{bc}\\) and
\\(\boxed{d}\\) is the midpoint of \\(\overline{ac}\\).
we can apply the \\(\boldsymbol{\text{triangle midsegment}}\\) theorem: \\(ed = \frac{1}{2}ba\\).
substituting in the values and solving for \\(x\\), we get \\(x = \\) value.
now, since \\(be = x\\), then \\(bc = \boxed{10}\\)

Explanation:

Step1: Identify the theorem

We use the triangle midsegment theorem, which states that the segment connecting the midpoints of two sides of a triangle is parallel to the third side and half its length. So \( ED=\frac{1}{2}BA \).

Step2: Substitute the expressions

Given \( ED = x + 2 \) and \( BA=4x - 6 \). By the midsegment theorem:
\( x + 2=\frac{1}{2}(4x - 6) \)

Step3: Solve for \( x \)

Multiply both sides by 2: \( 2(x + 2)=4x - 6 \)
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 \( BC \)

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

Answer:

The length of \( \overline{BC} \) is \( 10 \)