https://file.aiquickdraw.com/mathbot/file/1728185240385g08gcsyd.jpg

Let’s address the mathematical problem step-by-step from the image you provided.

### Given Points:
1. \( (-5, 6) \) and \( (2, -3) \)

### a) **Distance Between the Points:**
The formula for the distance between two points \((x_1, y_1)\) and \((x_2, y_2)\) is:
\[
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\]
Substitute the given points:
\[
d = \sqrt{(2 - (-5))^2 + (-3 - 6)^2} = \sqrt{(2 + 5)^2 + (-3 - 6)^2}
\]
\[
d = \sqrt{(7)^2 + (-9)^2} = \sqrt{49 + 81} = \sqrt{130}
\]

So, the correct distance is \( \sqrt{130} \), not \( \sqrt{81} \), which was incorrectly marked.

### b) **Midpoint of the Line Segment:**
The formula for the midpoint between two points is:
\[
\text{Midpoint} = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)
\]
Substitute the points:
\[
\text{Midpoint} = \left( \frac{-5 + 2}{2}, \frac{6 + (-3)}{2} \right) = \left( \frac{-3}{2}, \frac{3}{2} \right)
\]
The midpoint calculation is correct as shown on your paper.

### Summary:
1. Distance: Correct answer is \( \sqrt{130} \), not \( \sqrt{81} \).
2. Midpoint: Correctly calculated as \( \left( -\frac{3}{2}, \frac{3}{2} \right) \).

Would you like further clarification or more details on any part?

Here are 5 additional questions to explore:
1. Can you find the slope of the line passing through the two points?
2. How would you plot these points and the midpoint on a graph?
3. Can you use the distance formula for 3D points?
4. What are the coordinates of a point that is twice the distance away from \((-5, 6)\) along the line connecting the two points?
5. How would the midpoint change if one of the points was changed slightly?

**Tip:** The distance formula is derived from the Pythagorean theorem in 2D space!

Plot the points (-5, 6) and (2, -3). Then find: a) the distance between the points, b) the midpoint of the line segment joining the points.

Learn how to calculate the distance and midpoint between the points (-5, 6) and (2, -3). This solution uses the distance formula and midpoint formula, applying key concepts from coordinate geometry. Ideal for students in Grades 7-10.

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