Question

finding 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
\\(\boldsymbol{\text{option: c, d, e}}\\) is the midpoint of \\(\overline{ac}\\).
we apply the \\(\boldsymbol{\text{dropdown}}\\) theorem: \\(ed = \frac{1}{2}ba\\).
substituting in the expressions for the lengths and solving for \\(x\\), we get \\(x = \boldsymbol{\text{dropdown}}\\).
now, since \\(be = x\\), then \\(bc = \boldsymbol{10}\\).

diagram: triangle with vertices b, a, c. segment ba: \\(4x - 6\\). midpoints e (on bc) and d (on ac). segment ed: \\(x + 2\\). markings indicate midpoints and congruent segments.

Explanation:

Step1: Identify Midsegment Theorem

The problem involves a triangle with midpoints, so we use the Midsegment Theorem (also called Midline 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. Here, \( ED \) is the midsegment, so \( ED=\frac{1}{2}BA \).

Step2: Substitute Expressions

We know \( ED = x + 2 \) and \( BA=4x - 6 \). Substitute into the midsegment formula:
\( x + 2=\frac{1}{2}(4x - 6) \)

Step3: Solve for \( x \)

Multiply both sides by 2 to eliminate the fraction:
\( 2(x + 2)=4x - 6 \)
\( 2x + 4 = 4x - 6 \)
Subtract \( 2x \) from both sides:
\( 4 = 2x - 6 \)
Add 6 to both sides:
\( 10 = 2x \)
Divide by 2:
\( x = 5 \)

Step4: Find Length of \( BC \)

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

Answer:

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