Question

in the diagram below of triangle bcd, e is the midpoint of \\( \overline { b d } \\) and f is the midpoint of \\( \overline { c d } \\). if \\( e f = - 21 + 3 x \\), and \\( b c = - 24 + 4 x \\), what is the measure of \\( \overline { b c } \\)?

Explanation:

Step1: Use the mid - segment theorem

The mid - segment theorem states that the length of the mid - segment (\(EF\)) of a triangle is half the length of the third side (\(BC\)). So, \(BC = 2EF\).

Step2: Substitute the given expressions

Given \(EF=-21 + 3x\) and \(BC=-24 + 4x\), and since \(BC = 2EF\), we substitute:
\(-24 + 4x=2(-21 + 3x)\)

Step3: Solve the equation

Expand the right - hand side: \(-24 + 4x=-42+6x\).
Subtract \(4x\) from both sides: \(-24=-42 + 6x-4x\), which simplifies to \(-24=-42+2x\).
Add \(42\) to both sides: \(2x=-24 + 42\), so \(2x = 18\).
Divide both sides by \(2\): \(x = 9\).

Step4: Find the length of \(BC\)

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

Answer:

\(12\)