Point P is located on the segment between point  A(1,4)
and point  D(7,13).
The distance from A to P is twice the distance from P to D. What are the coordinates of point P?

To find the coordinates of point \( P \) on the line segment between points \( A(1,4) \) and \( D(7,13) \), given that the distance from \( A \) to \( P \) is twice the distance from \( P \) to \( D \), we can use the section formula. This formula allows us to find a point dividing a line segment into a given ratio.

### Step-by-Step Solution

1. **Identify the Coordinates and Ratio:**
   - Coordinates of \( A \): \( (1, 4) \)
   - Coordinates of \( D \): \( (7, 13) \)
   - The ratio in which \( P \) divides the segment \( AD \): \( 2:1 \). This means that \( AP:PD = 2:1 \).

2. **Use the Section Formula:**
   If a point \( P(x, y) \) divides a line segment joining two points \( A(x_1, y_1) \) and \( D(x_2, y_2) \) in the ratio \( m:n \), the coordinates of \( P \) can be found using the formula:

   \[
   P \left( \frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n} \right)
   \]

   Here, \( m = 2 \) and \( n = 1 \).

3. **Apply the Section Formula:**
   - For the \( x \)-coordinate of \( P \):
     \[
     x = \frac{(2 \times 7) + (1 \times 1)}{2+1} = \frac{14 + 1}{3} = \frac{15}{3} = 5
     \]
   - For the \( y \)-coordinate of \( P \):
     \[
     y = \frac{(2 \times 13) + (1 \times 4)}{2+1} = \frac{26 + 4}{3} = \frac{30}{3} = 10
     \]

4. **Conclusion:**
   The coordinates of point \( P \) are \( (5, 10) \).

### Final Answer
The coordinates of point \( P \) are \( \boxed{(5, 10)} \).

Would you like further details or have any questions?

### Related Questions:

1. What is the formula for finding the distance between two points in a plane?
2. How can you find the midpoint of a line segment given the coordinates of its endpoints?
3. If a point divides a line segment in a 3:2 ratio, how would you use the section formula to find that point?
4. How would the problem change if \( P \) were to divide the segment in the ratio \( 1:2 \) instead of \( 2:1 \)?
5. Can the section formula be used for three-dimensional space? If so, how?

**Tip:** The section formula is a powerful tool not only for finding a point dividing a line segment in a given ratio but also for solving many problems in coordinate geometry involving line segments and ratios.

Learn how to find the coordinates of point P on the line segment between points A(1,4) and D(7,13). The solution uses the section formula to divide the segment in a given ratio. Suitable for Grades 9-12.

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