Question

in the diagram below of triangle bcd, e is the midpoint of \\(\overline{bd}\\) and f is the midpoint of \\(\overline{cd}\\). if \\(ef = -21 + 3z\\), and \\(bc = -24 + 4z\\), what is the measure of \\(bc\\)?

Explanation:

Step1: Apply Midline Theorem

In triangle \(BCD\), \(E\) is the midpoint of \(BD\) and \(F\) is the midpoint of \(CD\), so by the Midline Theorem, \(EF=\frac{1}{2}BC\).

Step2: Substitute the Expressions

Substitute \(EF = -21 + 3z\) and \(BC = -24 + 4z\) into the equation \(EF=\frac{1}{2}BC\):
\[
-21 + 3z=\frac{1}{2}(-24 + 4z)
\]

Step3: Simplify the Equation

First, distribute the \(\frac{1}{2}\) on the right - hand side:
\[
-21 + 3z=-12 + 2z
\]
Then, subtract \(2z\) from both sides:
\[
-21+3z - 2z=-12 + 2z-2z
\]
\[
-21 + z=-12
\]
Next, add 21 to both sides:
\[
z=-12 + 21
\]
\[
z = 9
\]

Step4: Find the Measure of \(BC\)

Substitute \(z = 9\) into the expression for \(BC\): \(BC=-24 + 4z\)
\[
BC=-24+4\times9
\]
\[
BC=-24 + 36
\]
\[
BC = 12
\]

Answer:

\(12\)