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 = -9x + 66, and bc = 7 + 7x, what is the measure of \overline{ef}?

Explanation:

Step1: Apply the mid - segment theorem

The mid - segment theorem states that the length of the mid - segment (a segment connecting the midpoints of two sides of a triangle) is half the length of the third side. So, \(EF=\frac{1}{2}BC\).
Given \(EF = - 9x+66\) and \(BC = 7 + 7x\), we substitute into the formula: \(-9x + 66=\frac{1}{2}(7 + 7x)\).

Step2: Solve the equation for \(x\)

Multiply both sides of the equation \(-9x + 66=\frac{1}{2}(7 + 7x)\) by \(2\) to get rid of the fraction:
\(2(-9x + 66)=7 + 7x\).
Expand the left - hand side: \(-18x+132 = 7 + 7x\).
Add \(18x\) to both sides: \(132=7 + 7x+18x\).
Combine like terms: \(132=7 + 25x\).
Subtract \(7\) from both sides: \(132 - 7=25x\), so \(125 = 25x\).
Divide both sides by \(25\): \(x = 5\).

Step3: Find the length of \(EF\)

Substitute \(x = 5\) into the formula for \(EF\): \(EF=-9x + 66\).
\(EF=-9\times5+66\).
First, calculate \(-9\times5=-45\).
Then, \(EF=-45 + 66\).
\(EF = 21\).

Answer:

\(21\)