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 Midline Theorem

In triangle \( CDE \), \( F \) is the midpoint of \( \overline{CE} \) and \( G \) is the midpoint of \( \overline{DE} \). By the Midline Theorem (also known as the Midsegment Theorem), the segment connecting the midpoints of two sides of a triangle is parallel to the third side and half its length. So, \( FG=\frac{1}{2}CD \).

Step2: Set Up the Equation

We know \( FG = 5x - 25 \) and \( CD=9x - 41 \). Substituting into the midline theorem relationship:

$$ 5x - 25=\frac{1}{2}(9x - 41) $$

Step3: Solve for \( x \)

Multiply both sides of the equation by 2 to eliminate the fraction:

$$ 2(5x - 25)=9x - 41 $$

$$ 10x - 50 = 9x - 41 $$

Subtract \( 9x \) from both sides:

$$ 10x - 9x- 50=9x - 9x - 41 $$

$$ x - 50=- 41 $$

Add 50 to both sides:

$$ x-50 + 50=-41 + 50 $$

$$ x = 9 $$

Step4: Find the Length of \( CD \)

Substitute \( x = 9 \) into the expression for \( CD \): \( CD=9x - 41 \)

$$ CD=9(9)-41 $$

$$ CD = 81-41 $$

$$ CD = 40 $$

Answer:

\( 40 \)