Question

in the diagram below of triangle cde, f is the midpoint of \overline{ce} and g is the midpoint of \overline{de}. if fg = 5x - 25, and cd = 9x - 41, what is the measure of \overline{cd}?

Explanation:

Step1: Apply the mid - segment theorem

In a triangle, the mid - segment (a segment connecting the midpoints of two sides) is parallel to the third side and half its length. So, \(CD = 2FG\).

Step2: Substitute the given expressions

Since \(CD = 2FG\), we substitute \(FG=5x - 25\) and \(CD = 9x-41\) into the equation: \(9x-41=2(5x - 25)\).

Step3: Solve the equation for \(x\)

Expand the right - hand side: \(9x-41 = 10x-50\).
Subtract \(9x\) from both sides: \(-41=x - 50\).
Add 50 to both sides: \(x=9\).

Step4: Find the length of \(CD\)

Substitute \(x = 9\) into the expression for \(CD\): \(CD=9x-41\).
\(CD=9\times9-41\).
\(CD = 81-41\).
\(CD = 40\).

Answer:

\(40\)