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 solve this, we can use the concept of partitioning a line segment. Point \( P \) is located on the line segment between points \( A(1,4) \) and \( D(7,13) \), and the problem tells us that the distance from \( A \) to \( P \) is twice the distance from \( P \) to \( D \). This suggests a ratio of 2:1 for the distances \( AP : PD \).

### Step 1: Define the ratio
Given that \( AP : PD = 2 : 1 \), point \( P \) divides the line segment in the ratio \( 2:1 \). To find the coordinates of point \( P(x, y) \), we use the section formula for a point dividing a segment in a given ratio:

\[
P\left( x, y \right) = \left( \frac{m \cdot x_2 + n \cdot x_1}{m+n}, \frac{m \cdot y_2 + n \cdot y_1}{m+n} \right)
\]

where \( A(x_1, y_1) \) and \( D(x_2, y_2) \), and the ratio is \( m:n \).

### Step 2: Apply the coordinates and the ratio
Here, \( A(1, 4) \), \( D(7, 13) \), and the ratio is \( 2:1 \) (so \( m = 2 \) and \( n = 1 \)).

\[
P(x, y) = \left( \frac{2 \cdot 7 + 1 \cdot 1}{2 + 1}, \frac{2 \cdot 13 + 1 \cdot 4}{2 + 1} \right)
\]

### Step 3: Solve for \( P(x, y) \)
\[
x = \frac{2 \cdot 7 + 1 \cdot 1}{3} = \frac{14 + 1}{3} = \frac{15}{3} = 5
\]

\[
y = \frac{2 \cdot 13 + 1 \cdot 4}{3} = \frac{26 + 4}{3} = \frac{30}{3} = 10
\]

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

Would you like more details or have any questions about this solution?

Here are 5 related questions for practice:
1. What are the coordinates of a point dividing the segment between \( A(1, 2) \) and \( B(9, 8) \) in the ratio \( 3:2 \)?
2. If the point \( P \) divides a line segment in the ratio \( 1:3 \), how would the formula change?
3. How do you find the midpoint of a segment using the section formula?
4. Can the section formula be used in 3D? How?
5. How does the ratio of distances affect the location of a point on a segment?

**Tip**: The section formula is very useful when dealing with problems involving weighted averages of positions!

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?

Discover how to find the coordinates of point P on the segment between points A(1,4) and D(7,13) where the distance from A to P is twice that from P to D. This problem uses the section formula and is suitable for students in Grades 9-10.

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