https://file.aiquickdraw.com/mathbot/file/1730384817576twd8cp8g.jpg

This image shows three graphs with labeled points and line segments between them. I’ll go through each problem in the image and provide a description along with a general solution approach for each.

### Problem 2
**Points**: (-3, 0) and (4, 0)  
The line segment connects these two points along the x-axis.

- **Distance Calculation**: 
  Since both points are on the x-axis, the distance is the absolute difference in the x-coordinates:
  \[
  \text{Distance} = |4 - (-3)| = 4 + 3 = 7
  \]

- **Midpoint Calculation**:
  The midpoint between two points \((x_1, y_1)\) and \((x_2, y_2)\) is:
  \[
  \text{Midpoint} = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)
  \]
  Here:
  \[
  \text{Midpoint} = \left( \frac{-3 + 4}{2}, \frac{0 + 0}{2} \right) = \left( \frac{1}{2}, 0 \right)
  \]

### Problem 4
**Points**: (-4, -3) and (-4, 2)  
The line segment is vertical, connecting points with the same x-coordinate.

- **Distance Calculation**:
  Since the points share the same x-coordinate, the distance is the absolute difference in the y-coordinates:
  \[
  \text{Distance} = |2 - (-3)| = 2 + 3 = 5
  \]

- **Midpoint Calculation**:
  The midpoint between \((-4, -3)\) and \((-4, 2)\) is:
  \[
  \text{Midpoint} = \left( \frac{-4 + (-4)}{2}, \frac{-3 + 2}{2} \right) = \left( -4, -\frac{1}{2} \right)
  \]

### Problem 6
**Points**: (-2, 4) and (3, -5)  
This line segment is diagonal, connecting two points in different quadrants.

- **Distance Calculation**:
  Using the distance formula for two points \((x_1, y_1)\) and \((x_2, y_2)\):
  \[
  \text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
  \]
  Here:
  \[
  \text{Distance} = \sqrt{(3 - (-2))^2 + (-5 - 4)^2} = \sqrt{(3 + 2)^2 + (-9)^2} = \sqrt{5^2 + 9^2} = \sqrt{25 + 81} = \sqrt{106} \approx 10.3
  \]

- **Midpoint Calculation**:
  The midpoint between \((-2, 4)\) and \((3, -5)\) is:
  \[
  \text{Midpoint} = \left( \frac{-2 + 3}{2}, \frac{4 + (-5)}{2} \right) = \left( \frac{1}{2}, -\frac{1}{2} \right)
  \]

Let me know if you'd like additional details on any of these problems or further explanations!

### Follow-Up Questions
1. What is the formula for calculating the distance between two points in 3D space?
2. How would you find the slope of the line in Problem 6?
3. Can a vertical line segment have a slope? Why or why not?
4. What are the characteristics of points on the x-axis in a coordinate plane?
5. How can you verify if a point is the midpoint of a segment?

### Tip
Always visualize the points on a coordinate plane first to identify any shortcuts in distance or midpoint calculations.

Calculate the distance and midpoint between pairs of points in three different graphs.

This guide provides a step-by-step solution for calculating distances and midpoints between points on a coordinate plane. Ideal for students in Grades 8-10, covering the distance formula and midpoint formula for solving geometry problems.

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