Question

lm = x + 32 ln = 3x - 10 find mn
(diagram: line segment with points l, m, n (m between l and n), lm and mn each have one tick mark)

Explanation:

Step1: Determine the relationship of segments

From the diagram, \( M \) is the midpoint of \( LN \)? Wait, no, the marks show \( LM = MN \)? Wait, the tick marks: \( L \) to \( M \) and \( M \) to \( N \) have the same tick, so \( LM = MN \). Wait, but also \( LN = LM + MN \), and since \( LM = MN \), then \( LN = 2LM \).

So set \( LN = 2LM \):
\( 3x - 10 = 2(x + 32) \)

Step2: Solve for \( x \)

Expand right side: \( 3x - 10 = 2x + 64 \)
Subtract \( 2x \) from both sides: \( x - 10 = 64 \)
Add 10 to both sides: \( x = 74 \)

Step3: Find \( LM \) and \( MN \)

Since \( LM = MN \), first find \( LM \): \( LM = x + 32 = 74 + 32 = 106 \). So \( MN = LM = 106 \)? Wait, wait, maybe I misread. Wait, the diagram: \( L \)---M---N, with \( LM \) and \( MN \) having equal ticks, so \( LM = MN \), and \( LN = LM + MN = 2LM \). So the equation \( LN = 2LM \) is correct.

Wait, let's check again. \( LN = 3x - 10 \), \( LM = x + 32 \), and \( MN = LN - LM \)? Wait, no, if \( M \) is between \( L \) and \( N \), then \( LN = LM + MN \). But the tick marks: if \( LM \) and \( MN \) have the same tick, then \( LM = MN \). So \( LN = LM + LM = 2LM \). So that's the key.

So solving \( 3x - 10 = 2(x + 32) \):

\( 3x - 10 = 2x + 64 \)

\( 3x - 2x = 64 + 10 \)

\( x = 74 \)

Then \( LM = 74 + 32 = 106 \), \( MN = LN - LM \)? Wait, no, if \( LM = MN \), then \( MN = 106 \). Wait, but let's check \( LN \): \( 3(74) - 10 = 222 - 10 = 212 \). And \( LM + MN = 106 + 106 = 212 \), which matches. So \( MN = 106 \).

Answer:

\( 106 \)