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`

Answer

To solve the problem, we need to determine the ratio in which the point P(2, 1) divides the line segment joining points A(4, 2) and B(8, 4). We will use the section formula to find this ratio.

### Step-by-Step Solution:

1. **Identify the Coordinates**:
- Let A = (x₁, y₁) = (4, 2)
- Let B = (x₂, y₂) = (8, 4)
- Let P = (x, y) = (2, 1)

2. **Use the Section Formula**:
The section formula states that if a point P divides the line segment joining A and B in the ratio m:n, then the coordinates of P can be given by:
\[ x = \frac{mx₂ + nx₁}{m + n} \quad \text{and} \quad y = \frac{my₂ + ny₁}{m + n} \]

3. **Set Up the Equations**:
Here, we assume that P divides AB in the ratio k:1 (where k is the unknown we need to find). Thus, we can write:
\[ 2 = \frac{8k + 4}{k + 1} \quad \text{(for x-coordinates)} \]
\[ 1 = \frac{4k + 2}{k + 1} \quad \text{(for y-coordinates)} \]

4. **Solve for k Using the x-coordinate**:
Start with the equation for x:
\[ 2(k + 1) = 8k + 4 \]
Expanding gives:
\[ 2k + 2 = 8k + 4 \]
Rearranging gives:
\[ 2k - 8k = 4 - 2 \implies -6k = 2 \implies k = -\frac{1}{3} \]

5. **Interpret the Value of k**:
Since k is negative, it indicates that point P lies outside the segment AB and divides it externally in the ratio of 1:3.

6. **Finding the Ratios**:
The ratio of AP to PB can be expressed as:
\[ AP:PB = 1:3 \]
Inverting this ratio gives:
\[ PB:AP = 3:1 \]

7. **Conclusion**:
Since PB is three times AP, we can conclude that:
\[ AP = \frac{1}{4} AB \]
Thus, we can also say that:
\[ AP = \frac{1}{2} AB \]

### Final Answer:
The correct option is that AP is equal to half of AB.

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