Question

m is the midpoint of (overline{ln}). (lm = 2x - 2) (mn = x + 18) find (ln) (diagram: points l, m, n on a line with m between l and n)

Explanation:

Step1: Use midpoint property

Since M is the midpoint of \( \overline{LN} \), \( LM = MN \). So we set up the equation:
\( 2x - 2 = x + 18 \)

Step2: Solve for x

Subtract \( x \) from both sides:
\( 2x - x - 2 = 18 \)
\( x - 2 = 18 \)
Add 2 to both sides:
\( x = 18 + 2 \)
\( x = 20 \)

Step3: Find LM and MN

First, find \( LM \): \( LM = 2x - 2 = 2(20) - 2 = 40 - 2 = 38 \)
Then, find \( MN \): \( MN = x + 18 = 20 + 18 = 38 \) (which checks out since \( LM = MN \))

Step4: Find LN

Since \( LN = LM + MN \), substitute the values:
\( LN = 38 + 38 = 76 \)

Answer:

\( 76 \)