Question

look at the diagram and evaluate. which of the statements below is false? answer options: mo > lo; n is the midpoint of \\(\overline{mo}\\); lm > no; ln > lm

Explanation:

First, we identify the positions of each point:
- \( L \) is at 1, \( M \) is at 8, \( N \) is at 11, \( O \) is at 14.

Step 1: Calculate \( MO \), \( LO \), \( LM \), \( NO \), \( LN \)

- \( MO = |14 - 8| = 6 \)
- \( LO = |14 - 1| = 13 \) Wait, no, wait: Wait, \( MO \) is from \( M(8) \) to \( O(14) \), so \( MO = 14 - 8 = 6 \). \( LO \) is from \( L(1) \) to \( O(14) \), so \( LO = 14 - 1 = 13 \)? Wait, no, that can't be. Wait, maybe I mixed up. Wait, \( MO \): \( M \) is 8, \( O \) is 14, so \( MO = 14 - 8 = 6 \). \( LO \): \( L \) is 1, \( O \) is 14, so \( LO = 14 - 1 = 13 \). But then \( MO > LO \) would be 6 > 13? No, that's not right. Wait, maybe I made a mistake. Wait, no, maybe the points are: \( L \) at 1, \( M \) at 8, \( N \) at 11, \( O \) at 14. Let's recalculate:

- \( LM \): distance from \( L(1) \) to \( M(8) \): \( 8 - 1 = 7 \)
- \( NO \): distance from \( N(11) \) to \( O(14) \): \( 14 - 11 = 3 \)
- \( LN \): distance from \( L(1) \) to \( N(11) \): \( 11 - 1 = 10 \)
- \( MO \): distance from \( M(8) \) to \( O(14) \): \( 14 - 8 = 6 \)
- \( LO \): distance from \( L(1) \) to \( O(14) \): \( 14 - 1 = 13 \)
- Midpoint of \( MO \): \( M(8) \) and \( O(14) \), midpoint is \( \frac{8 + 14}{2} = 11 \), which is \( N \). So \( N \) is the midpoint of \( MO \).

Now check each statement:

1. \( MO > LO \): \( MO = 6 \), \( LO = 13 \). 6 > 13? False? Wait, no, maybe I messed up the direction. Wait, no, distance is absolute. Wait, maybe the problem is that \( MO \) is 6, \( LO \) is 13, so \( MO > LO \) is false? But let's check other options.

2. \( N \) is the midpoint of \( \overline{MO} \): midpoint of 8 and 14 is \( (8 + 14)/2 = 11 \), which is \( N \). So this is true.

3. \( LM > NO \): \( LM = 7 \), \( NO = 3 \). 7 > 3: true.

4. \( LN > LM \): \( LN = 10 \), \( LM = 7 \). 10 > 7: true.

Wait, but the first option: \( MO > LO \). \( MO = 6 \), \( LO = 13 \). So 6 > 13 is false. But wait, maybe I made a mistake in the positions. Wait, the diagram: \( L \) is at 1, \( M \) at 8, \( N \) at 11, \( O \) at 14. So:

- \( MO \): 14 - 8 = 6
- \( LO \): 14 - 1 = 13
- So \( MO > LO \) is 6 > 13: false. But wait, the other options:

Wait, maybe the question is which is FALSE. Let's check again.

Wait, maybe I mixed up \( MO \) and \( LO \). Wait, \( LO \) is from \( L \) to \( O \): 14 - 1 = 13. \( MO \) is from \( M \) to \( O \): 14 - 8 = 6. So \( MO > LO \) is false. But let's check the other options.

Wait, the options are:

- \( MO > LO \)
- \( N \) is the midpoint of \( \overline{MO} \)
- \( LM > NO \)
- \( LN > LM \)

Wait, but maybe I made a mistake in \( LM \). \( LM \) is from \( L(1) \) to \( M(8) \): 8 - 1 = 7. \( NO \) is from \( N(11) \) to \( O(14) \): 14 - 11 = 3. So 7 > 3: true. \( LN \) is from \( L(1) \) to \( N(11) \): 11 - 1 = 10. \( LM \) is 7: 10 > 7: true. \( N \) is midpoint of \( MO \): (8 + 14)/2 = 11, which is \( N \): true. So the false one is \( MO > LO \)? Wait, but that seems counterintuitive. Wait, maybe the diagram is different. Wait, maybe \( L \) is at 0? No, the diagram shows \( L \) at 1, \( M \) at 8, \( N \) at 11, \( O \) at 14.

Wait, maybe the problem is that \( MO \) is 6, \( LO \) is 13, so \( MO > LO \) is false. So the answer is \( MO > LO \). But wait, let's check again.

Wait, maybe I misread the positions. Let's look at the number line:

- \( L \) is at 1 (between 0 and 2)
- \( M \) is at 8 (between 7 and 9)
- \( N \) is at 11 (between 10 and 12)
- \( O \) is at 14 (between 13 and 15)

So:

- \( LM \): 8 - 1 = 7
- \( LN \): 11 - 1 = 10
- \( NO \): 14 - 11 = 3
- \( MO \): 14 - 8 = 6
- \( LO \): 14 - 1 = 13

Now check each statement:

1. \( MO > LO \): 6 > 13? No, false.
2. \( N \) is midpoint of \( MO \): (8 + 14)/2 = 11, which is \( N \): true.
3. \( LM > NO \): 7 > 3: true.
4. \( LN > LM \): 10 > 7: true.

So the false statement is \( MO > LO \). Wait, but the option is \( MO > LO \). So that's the false one.

Answer:

\( MO > LO \)