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Calculate point P on AB¯¯¯¯¯¯¯¯ such that the ratio of AP to PB is 1:3.

To solve for point \( P \) on the line segment \( \overline{AB} \) such that the ratio of \( AP \) to \( PB \) is \( 1:3 \), follow these steps:

### Step 1: Determine Coordinates of Points \( A \) and \( B \)
From the graph:
- \( A \) is at \((-3, -4)\).
- \( B \) is at \((2, 3)\).

### Step 2: Use the Section Formula
The section formula finds a point dividing a line segment into a given ratio:
\[
P(x, y) = \left(\frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n}\right)
\]
where:
- \( (x_1, y_1) \) are coordinates of \( A \),
- \( (x_2, y_2) \) are coordinates of \( B \),
- \( m:n \) is the given ratio.

### Step 3: Apply the Ratio \( 1:3 \)
Let \( m = 1 \) and \( n = 3 \). Thus, \( P \) divides \( \overline{AB} \) in the ratio \( 1:3 \).

### Step 4: Substitute the Values
Substitute into the formula:
\[
x = \frac{(1)(2) + (3)(-3)}{1+3} = \frac{2 - 9}{4} = \frac{-7}{4} = -1.75
\]
\[
y = \frac{(1)(3) + (3)(-4)}{1+3} = \frac{3 - 12}{4} = \frac{-9}{4} = -2.25
\]

### Step 5: Final Answer
The coordinates of point \( P \) are:
\[
P = \left(-1.75, -2.25\right)
\]

Would you like further details or have any questions?

### Related Questions
1. How can you find the coordinates of a point that divides a segment in any given ratio?
2. What is the formula for finding the midpoint of a line segment?
3. How do you derive the section formula?
4. What changes in the formula if the ratio is reversed (e.g., 3:1)?
5. How do you find the length of a line segment between two points?

### Tip
Always verify the coordinates by checking the ratio visually or using a tool, especially when working with graphical representations!

Calculate point P on AB such that the ratio of AP to PB is 1:3.

Learn how to calculate the coordinates of point P on the line segment AB such that AP to PB is in the ratio 1:3. This problem covers key concepts in coordinate geometry, including the section formula and ratios. Suitable for students in Grades 9-11.

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