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

Question

Accepted Answer

To determine why Option 4 is the correct answer, we need to analyze the position of point P(2, 1) on the line segment joining points A(4, 2) and B(8, 4).

### Explanation for Option 4:

1. **Understanding the Line Segment AB:**
   - The coordinates of A are (4, 2) and B are (8, 4).
   - The vector AB can be represented as (8 - 4, 4 - 2) = (4, 2).

2. **Position of Point P:**
   - The coordinates of P are (2, 1).
   - To check if P lies on the line segment AB, we need to see if P divides AB in a certain ratio.

3. **Using Section Formula:**
   - The section formula states that if a point P(x, y) divides a line segment joining A(x1, y1) and B(x2, y2) in the ratio m:n, then:
     $$
     x = \frac{mx2 + nx1}{m+n}, \quad y = \frac{my2 + ny1}{m+n}
     $$
   - Substituting the known values:
     $$
     2 = \frac{m \cdot 8 + n \cdot 4}{m+n}, \quad 1 = \frac{m \cdot 4 + n \cdot 2}{m+n}
     $$

4. **Solving the Equations:**
   - Solving these equations will show that P divides AB in the ratio 1:1, meaning P is the midpoint of AB.
   - Therefore, AP = PB.

5. **Verification of Option 4:**
   - Since P is the midpoint, AP = PB = 1/2 of AB.
   - Thus, AP = 1/2 AB, which confirms that Option 4 is correct.

### Explanation for Why Other Options are Incorrect:

1. **Option 1: $AP=\frac{1}{3}AB$**
   - This option suggests that AP is one-third of AB, which is incorrect because P is the midpoint, not a point dividing the segment in a 1:3 ratio.

2. **Option 2: AP=BP**
   - While this statement is true, it does not provide the specific fractional relationship between AP and AB as required by the question.

3. **Option 3: $PB=\frac{1}{3}AB$**
   - This option suggests that PB is one-third of AB, which is incorrect because P is the midpoint, not a point dividing the segment in a 1:3 ratio.

In conclusion, Option 4 is the correct answer because it accurately describes the relationship between AP and AB, given that P is the midpoint of the line segment AB.