Question

if the length of \\( \overline { b e } \\) is \\( 2 x - 9 \\) and the length of \\( \overline { c d } \\) is \\( 7 x - 42 \\), what is the length of \\( \overline { c d } \\)? your answer should be a number and not contain any variables.

Explanation:

Step1: Identify the midline property

From the diagram, \( \overline{BE} \) is a midline segment (since the marks indicate equal segments), so \( BE=\frac{1}{2}CD \). So we set up the equation \( 2x - 9=\frac{1}{2}(7x - 42) \).

Step2: Solve for \( x \)

Multiply both sides by 2 to eliminate the fraction: \( 2(2x - 9)=7x - 42 \)
Expand the left side: \( 4x-18 = 7x - 42 \)
Subtract \( 4x \) from both sides: \( - 18=3x - 42 \)
Add 42 to both sides: \( 24 = 3x \)
Divide both sides by 3: \( x = 8 \)

Step3: Find the length of \( \overline{CD} \)

Substitute \( x = 8 \) into the expression for \( CD \): \( CD=7x - 42=7(8)-42 \)
Calculate \( 7\times8 = 56 \), then \( 56 - 42=14 \)

Answer:

14