finding an unknown length
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 = $

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# Explanation:
## Step1: Identify the mid - point
From the markings on the diagram, we can tell $E$ is the midpoint of $\overline{BC}$ and $D$ is the midpoint of $\overline{AC}$.
## Step2: Apply the mid - segment theorem
We can apply the mid - segment theorem: $ED=\frac{1}{2}BA$.
## Step3: Substitute and solve for $x$
Substitute $ED = x + 2$ and $BA=4x - 6$ into $ED=\frac{1}{2}BA$.
$$
\begin{align*}
x + 2&=\frac{1}{2}(4x - 6)\
x+2&=2x-3\
2 + 3&=2x-x\
x&=5
\end{align*}
$$
## Step4: Calculate the length of $BC$
Since $BE=x$ and $BC = 2BE$ (because $E$ is the midpoint of $BC$), then $BC = 2x$. Substitute $x = 5$ into $BC = 2x$, so $BC=10$.

# Answer:
$D$ is the midpoint of $\overline{AC}$; mid - segment; $x = 5$; $BC = 10$

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QUESTION IMAGE

Question

Explanation:

Step1: Identify the mid - point

Step2: Apply the mid - segment theorem

Step3: Substitute and solve for \(x\)

Step4: Calculate the length of \(BC\)

Answer: