If the point P(2,1)lies on the line segment joining points A(4,2)and B(8,4), then

Options

A. AP `=(1)/(3)AB`
B. `AP=PB`
C. PB `=(1)/(3)AB`
D. `AP=(1)/(2)AB`

Explanation

To solve the problem, we need to determine the relationship between the distances AP, PB, and AB where P(2,1) lies on the line segment joining points A(4,2) and B(8,4).

### Step-by-Step Solution:

1. **Identify the Points:**
- Point A = (4, 2)
- Point B = (8, 4)
- Point P = (2, 1)

2. **Use the Distance Formula:**
The distance formula between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is given by:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]

3. **Calculate Distance AP:**
- For points A(4, 2) and P(2, 1):
\[ AP = \sqrt{(2 - 4)^2 + (1 - 2)^2} = \sqrt{(-2)^2 + (-1)^2} = \sqrt{4 + 1} = \sqrt{5} \]

4. **Calculate Distance PB:**
- For points P(2, 1) and B(8, 4):
\[ PB = \sqrt{(8 - 2)^2 + (4 - 1)^2} = \sqrt{(6)^2 + (3)^2} = \sqrt{36 + 9} = \sqrt{45} = 3\sqrt{5} \]

5. **Calculate Distance AB:**
- For points A(4, 2) and B(8, 4):
\[ AB = \sqrt{(8 - 4)^2 + (4 - 2)^2} = \sqrt{(4)^2 + (2)^2} = \sqrt{16 + 4} = \sqrt{20} = 2\sqrt{5} \]

6. **Establish Relationships:**
- We found:
- \( AP = \sqrt{5} \)
- \( PB = 3\sqrt{5} \)
- \( AB = 2\sqrt{5} \)

- Now, we can express the relationships:
- \( AP + PB = AB \)
- \( \sqrt{5} + 3\sqrt{5} = 4\sqrt{5} \) (which is not equal to \( 2\sqrt{5} \))
- However, we can find that \( PB = 3 \times AP \) and \( AB = 2 \times AP \).

7. **Final Conclusion:**
- Since \( AP = \frac{1}{2} AB \), we can conclude that the correct relationship is:
- \( AP = \frac{1}{2} AB \)

### Answer:
The correct option is that \( AP = \frac{1}{2} AB \).