Question

Points L, M, and N are collinear.
You are given LM = 13 and LN = 20. What is a possible value of MN?

Answer 1

Since points L, M, and N are collinear, we can position them on a line. Depending on their arrangement, the value of \( MN \) can vary.

Given:
- \( LM = 13 \)
- \( LN = 20 \)

We consider two possible arrangements of the points:

1. **Case 1:** The points are arranged as \( L - M - N \).

In this case, \( LN = LM + MN \). Therefore,
\[
20 = 13 + MN
\]
Solving for \( MN \):
\[
MN = 20 - 13 = 7
\]

2. **Case 2:** The points are arranged as \( L - N - M \).

In this case, \( LN = LM - MN \). Therefore,
\[
20 = 13 - MN
\]
Solving for \( MN \):
\[
MN = 13 - 20 = -7
\]
Since distances cannot be negative, this arrangement does not give a valid solution.

3. **Case 3:** The points are arranged as \( M - L - N \).

In this case, \( LN = MN + LM \). Therefore,
\[
20 = MN + 13
\]
Solving for \( MN \):
\[
MN = 20 - 13 = 7
\]

4. **Case 4:** The points are arranged as \( N - L - M \).

In this scenario, \( LN = MN - LM \). Therefore,
\[
20 = MN - 13
\]
Solving for \( MN \):
\[
MN = 20 + 13 = 33
\]

Thus, based on cases 1, 3, and 4, the possible values of \( MN \) that can be determined are:

- From Case 1 and Case 3: \( MN = 7 \)
- From Case 4: \( MN = 33 \)

Therefore, a possible value of \( MN \) is **7** or **33**.