5. Plotting, Distance, and Midpoint Plot the 
points (−2, 6) and (4, −3). Then find the distance 
between the points and the midpoint of the line segment 
joining the points

Let's break down the problem step by step.

### 1. Plotting the Points
First, let's plot the points \((-2, 6)\) and \((4, -3)\) on a Cartesian plane.

- \((-2, 6)\): This point is located 2 units to the left of the origin and 6 units up.
- \((4, -3)\): This point is located 4 units to the right of the origin and 3 units down.

### 2. Finding the Distance Between the Points
The distance \(d\) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by the distance formula:
\[
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\]

Substitute the coordinates of the given points into the formula:

\[
d = \sqrt{(4 - (-2))^2 + (-3 - 6)^2} = \sqrt{(4 + 2)^2 + (-3 - 6)^2} = \sqrt{6^2 + (-9)^2} = \sqrt{36 + 81} = \sqrt{117} \approx 10.82
\]

### 3. Finding the Midpoint
The midpoint \(M\) of the line segment joining two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by the midpoint formula:
\[
M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)
\]

Substitute the coordinates of the given points into the formula:

\[
M = \left(\frac{-2 + 4}{2}, \frac{6 + (-3)}{2}\right) = \left(\frac{2}{2}, \frac{3}{2}\right) = (1, 1.5)
\]

### Summary of Results
- **Distance between the points**: \(\sqrt{117} \approx 10.82\)
- **Midpoint**: \((1, 1.5)\)

Would you like the plotted graph of these points or any further details? 

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### Related Questions:
1. How do you derive the distance formula from the Pythagorean theorem?
2. What is the significance of the midpoint in a line segment?
3. How would the midpoint change if the points were reflected over the y-axis?
4. Can you find the equation of the line passing through the points \((-2, 6)\) and \((4, -3)\)?
5. What are the implications if the distance between two points is zero?

### Tip:
When finding the distance between points, always double-check your subtraction to avoid sign errors!

Learn how to plot points (-2, 6) and (4, -3), find the distance between them using the distance formula, and determine their midpoint with the midpoint formula. Detailed step-by-step solution included. Suitable for students in Grades 9-10.

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