Question

m is the midpoint of \\(\overline{ln}\\). if \\(lm = 9x\\) and \\(mn = x + 8\\), what is \\(mn\\)? simplify your answer and write it as a proper fraction, mixed number, or integer.

Explanation:

Step1: Use midpoint property

Since \( M \) is the midpoint of \( \overline{LN} \), \( LM = MN \). So we set up the equation \( 9x = x + 8 \).

Step2: Solve for \( x \)

Subtract \( x \) from both sides: \( 9x - x = x + 8 - x \), which simplifies to \( 8x = 8 \). Then divide both sides by 8: \( \frac{8x}{8}=\frac{8}{8} \), so \( x = 1 \).

Step3: Find \( MN \)

Substitute \( x = 1 \) into the expression for \( MN \), which is \( x + 8 \). So \( MN = 1 + 8 = 9 \).

Answer:

9