To determine the relationship of point P(4,2) with respect to the line segment joining points A(2,1) and B(8,4), we can use the section formula. The section formula helps us find the coordinates of a point that divides a line segment into a specific ratio.
### Step-by-Step Solution:
1. **Identify the Points:**
- Let A = (2, 1) and B = (8, 4).
- Let P = (4, 2).
2. **Assume the Ratio:**
- Assume point P divides the line segment AB in the ratio k:1, where k is an unknown value.
3. **Apply the Section Formula:**
- The coordinates of point P can be expressed using the section formula:
\[
P(x, y) = \left( \frac{m \cdot x_2 + n \cdot x_1}{m+n}, \frac{m \cdot y_2 + n \cdot y_1}{m+n} \right)
\]
- Here, \( m = k \), \( n = 1 \), \( x_1 = 2 \), \( y_1 = 1 \), \( x_2 = 8 \), and \( y_2 = 4 \).
4. **Set Up the Equations:**
- For the x-coordinate:
\[
4 = \frac{k \cdot 8 + 1 \cdot 2}{k + 1}
\]
- For the y-coordinate:
\[
2 = \frac{k \cdot 4 + 1 \cdot 1}{k + 1}
\]
5. **Solve the x-coordinate Equation:**
- Cross-multiply:
\[
4(k + 1) = 8k + 2
\]
- Expand and rearrange:
\[
4k + 4 = 8k + 2 \implies 4 = 8k - 4k \implies 4 = 4k \implies k = 1
\]
6. **Solve the y-coordinate Equation:**
- Cross-multiply:
\[
2(k + 1) = 4k + 1
\]
- Expand and rearrange:
\[
2k + 2 = 4k + 1 \implies 2 = 4k - 2k \implies 2 = 2k \implies k = 1
\]
7. **Conclusion:**
- Since both calculations give us \( k = 1 \), point P divides the line segment AB in the ratio 1:1.
### Final Answer:
Point P(4,2) divides the line segment joining points A(2,1) and B(8,4) in the ratio 1:1.
