Question

in the diagram below of triangle bcd, e is the midpoint of bd and f is the midpoint of cd. if ef = -6x + 35, and bc = 9x - 35, what is the measure of ef?

Explanation:

Step1: Use the mid - segment theorem

In a triangle, the mid - segment (a segment connecting the midpoints of two sides of a triangle) is parallel to the third side and half its length. So, \(EF=\frac{1}{2}BC\).
Given \(EF = - 6x+35\) and \(BC=9x - 35\). Then \(-6x + 35=\frac{1}{2}(9x - 35)\).

Step2: Solve the equation for \(x\)

Multiply both sides of the equation \(-6x + 35=\frac{1}{2}(9x - 35)\) by \(2\) to get rid of the fraction:
\(2(-6x + 35)=9x-35\).
Expand the left - hand side: \(-12x+70 = 9x-35\).
Add \(12x\) to both sides: \(70=9x + 12x-35\).
Combine like terms: \(70 = 21x-35\).
Add \(35\) to both sides: \(70 + 35=21x\), so \(105 = 21x\).
Divide both sides by \(21\): \(x = 5\).

Step3: Find the length of \(EF\)

Substitute \(x = 5\) into the formula for \(EF\): \(EF=-6x + 35\).
\(EF=-6\times5+35\).
\(EF=-30 + 35\).

Answer:

\(5\)