A line contains the points A,B,C and D. Point B is between points A and C. Point D is between points C and B. Which of the following inequalities must be true about lengths of these segments?

Options

A. BC lt AB
B. BD lt AB
C. BD lt CD
D. CD lt BC

Answer

To solve the problem, we need to analyze the positions of points A, B, C, and D on a line and determine which inequalities must be true based on their relative positions.

### Step-by-Step Solution:

1. **Understanding the Positioning of Points**:
- We know that point B is between points A and C. This means that the order of points on the line is A, B, C.
- Point D is between points C and B. This means that the order of points on the line is C, D, B.

2. **Visual Representation**:
- We can represent the points on a number line for clarity:
```
A ---- B ---- C
|
D
```
- Here, D is positioned between C and B.

3. **Identifying the Lengths**:
- Let's denote the lengths of segments as follows:
- Length of segment AB = |AB|
- Length of segment BC = |BC|
- Length of segment CD = |CD|
- Length of segment BD = |BD|

4. **Analyzing the Inequalities**:
- Since D is between C and B, we can conclude that:
- The length of segment CD (|CD|) must be less than the length of segment BC (|BC|) because D is a point that lies within the segment BC.
- Therefore, we can express this as:
\[ |CD| < |BC| \]

5. **Conclusion**:
- The only inequality that must be true based on the positions of points A, B, C, and D is:
\[ CD < BC \]
- Hence, the correct answer is that the inequality \( CD < BC \) must always hold true.