Question

1. points l, m, and n are collinear. point m bisects ln. find ln if lm = 3x - 10; mn = 2x + 20

Explanation:

Step1: Use the midpoint property

Since M bisects LN, \( LM = MN \). So we set the two expressions equal: \( 3x - 10 = 2x + 20 \).

Step2: Solve for x

Subtract \( 2x \) from both sides: \( 3x - 2x - 10 = 20 \), which simplifies to \( x - 10 = 20 \). Then add 10 to both sides: \( x = 20 + 10 = 30 \).

Step3: Find LM and MN

First, find \( LM \): substitute \( x = 30 \) into \( LM = 3x - 10 \), so \( LM = 3(30) - 10 = 90 - 10 = 80 \).
Then, find \( MN \): substitute \( x = 30 \) into \( MN = 2x + 20 \), so \( MN = 2(30) + 20 = 60 + 20 = 80 \).

Step4: Find LN

Since \( LN = LM + MN \), and \( LM = MN = 80 \), then \( LN = 80 + 80 = 160 \).

Answer:

160