Question

if the length of $\overline{be}$ is $3x - 15$ and the length of $\overline{cd}$ is $9x - 51$, what is the length of $\overline{be}$? your answer should be a number and not contain any variables.

Explanation:

Step1: Identify Midsegment Theorem

From the diagram, \( BE \) is a midsegment of triangle \( ACD \) (since \( B \) and \( E \) are midpoints, as indicated by the tick marks). By the Midsegment Theorem, the length of the midsegment is half the length of the third side. So, \( BE=\frac{1}{2}CD \).

Step2: Set up the equation

Given \( BE = 3x - 15 \) and \( CD=9x - 51 \), substitute into the midsegment formula:
\( 3x - 15=\frac{1}{2}(9x - 51) \)

Step3: Solve for \( x \)

Multiply both sides by 2 to eliminate the fraction:
\( 2(3x - 15)=9x - 51 \)
\( 6x - 30 = 9x - 51 \)

Subtract \( 6x \) from both sides:
\( - 30=3x - 51 \)

Add 51 to both sides:
\( 21 = 3x \)

Divide both sides by 3:
\( x = 7 \)

Step4: Find length of \( BE \)

Substitute \( x = 7 \) into \( BE = 3x - 15 \):
\( BE=3(7)-15 \)
\( BE = 21 - 15 \)
\( BE = 6 \)

Answer:

6