Line segment AB where A(-2, 2) and B(3, -3)

To analyze line segment \( AB \) where \( A(-2, 2) \) and \( B(3, -3) \), we can calculate the following:

### 1. **Slope of the Line Segment \( AB \):**
The slope \( m \) of the line segment connecting points \( A(x_1, y_1) \) and \( B(x_2, y_2) \) is given by:

\[
m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{-3 - 2}{3 - (-2)} = \frac{-5}{5} = -1
\]

### 2. **Midpoint of Line Segment \( AB \):**
The midpoint \( M \) of the line segment is calculated as:

\[
M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) = \left( \frac{-2 + 3}{2}, \frac{2 - 3}{2} \right) = \left( \frac{1}{2}, \frac{-1}{2} \right)
\]

### 3. **Length of Line Segment \( AB \):**
The length \( d \) of the line segment can be found using the distance formula:

\[
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} = \sqrt{(3 - (-2))^2 + (-3 - 2)^2} = \sqrt{5^2 + (-5)^2} = \sqrt{25 + 25} = \sqrt{50} = 5\sqrt{2}
\]

### Summary:
- **Slope**: \( -1 \)
- **Midpoint**: \( \left( \frac{1}{2}, \frac{-1}{2} \right) \)
- **Length**: \( 5\sqrt{2} \)

Would you like further details on any of these points?

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Here are 5 related questions:
1. How do you find the equation of the line passing through points \( A \) and \( B \)?
2. What is the equation of the perpendicular bisector of segment \( AB \)?
3. How do you calculate the area of a triangle formed by points \( A \), \( B \), and a third point \( C \)?
4. How can you determine if another point \( C(x_3, y_3) \) lies on the line segment \( AB \)?
5. What are the coordinates of a point \( P \) that divides the segment \( AB \) in a given ratio?

**Tip**: When calculating the distance between two points, remember that the distance formula is derived from the Pythagorean theorem applied in the coordinate plane.

Explore the analysis of line segment AB from point A(-2, 2) to point B(3, -3) using slope calculation, midpoint determination, and length calculation. Understand how to apply the slope formula, midpoint formula, and distance formula in coordinate geometry. Suitable for students in Grades 9-12.

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