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 + 3x, and bc = -24 + 4x, what is the measure of \overline{bc}?

Explanation:

Step1: Use the mid - segment theorem

In \(\triangle BCD\), since \(E\) is the mid - point of \(\overline{BD}\) and \(F\) is the mid - point of \(\overline{CD}\), by the mid - segment theorem, \(EF=\frac{1}{2}BC\).

Step2: Substitute the given expressions

Given \(EF = - 21+3x\) and \(BC=-24 + 4x\). Substitute into \(EF=\frac{1}{2}BC\), we get \(-21 + 3x=\frac{1}{2}(-24 + 4x)\).

Step3: Solve the equation for \(x\)

Multiply both sides of the equation \(-21 + 3x=\frac{1}{2}(-24 + 4x)\) by \(2\) to get \(2(-21 + 3x)=-24 + 4x\).
Expand: \(-42+6x=-24 + 4x\).
Subtract \(4x\) from both sides: \(-42+6x-4x=-24 + 4x-4x\), which simplifies to \(-42 + 2x=-24\).
Add \(42\) to both sides: \(-42+2x + 42=-24 + 42\), so \(2x=18\).
Divide both sides by \(2\): \(x = 9\).

Step4: Find the length of \(BC\)

Substitute \(x = 9\) into the expression for \(BC\), \(BC=-24 + 4x\).
\(BC=-24+4\times9\).
\(BC=-24 + 36\).
\(BC = 12\).

Answer:

\(12\)