Question

c is the midpoint of \\(\overline{bd}\\). if \\(cd = 9x - 10\\) and \\(bd = 15x - 4\\), what is \\(cd\\)? simplify your answer and write it as a proper fraction, mixed number, or integer.

Explanation:

Step1: Use midpoint property

Since \( C \) is the midpoint of \( \overline{BD} \), we have \( BD = 2 \times CD \).
Substitute \( CD = 9x - 10 \) and \( BD = 15x - 4 \) into the equation:
\( 15x - 4 = 2(9x - 10) \)

Step2: Solve for \( x \)

Expand the right - hand side: \( 15x - 4 = 18x - 20 \)
Subtract \( 15x \) from both sides: \( - 4=3x - 20 \)
Add 20 to both sides: \( 16 = 3x \)
Divide both sides by 3: \( x=\frac{16}{3} \)

Step3: Find the value of \( CD \)

Substitute \( x = \frac{16}{3} \) into the expression for \( CD \):
\( CD=9x - 10=9\times\frac{16}{3}-10 \)
First, calculate \( 9\times\frac{16}{3}=48 \)
Then, \( CD = 48-10 = 38 \)

Answer:

\( 38 \)