https://file.aiquickdraw.com/mathbot/file/1726626797815d24m030a.jpg

Let's solve the problem based on the graph and instructions provided:

### (a) Finding the length of \( AB \)
The formula for the distance between two points \( A(x_1, y_1) \) and \( B(x_2, y_2) \) is:
\[
AB = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\]
From the graph, the coordinates of \( A \) and \( B \) are:
- \( A(-6, 2) \)
- \( B(3, -1) \)

Substitute these into the distance formula:
\[
AB = \sqrt{(3 - (-6))^2 + (-1 - 2)^2}
\]
\[
AB = \sqrt{(3 + 6)^2 + (-1 - 2)^2} = \sqrt{9^2 + (-3)^2} = \sqrt{81 + 9} = \sqrt{90}
\]
\[
AB \approx 9.5
\]
So, \( AB \) is approximately **9.5** units.

### (b) Finding the midpoint of \( AB \)
The formula for the midpoint between two points \( A(x_1, y_1) \) and \( B(x_2, y_2) \) is:
\[
M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)
\]
Substitute the coordinates of \( A(-6, 2) \) and \( B(3, -1) \):
\[
M = \left( \frac{-6 + 3}{2}, \frac{2 + (-1)}{2} \right) = \left( \frac{-3}{2}, \frac{1}{2} \right) = (-1.5, 0.5)
\]
So, the midpoint is **(-1.5, 0.5)**.

### Summary:
- (a) The length of \( AB \) is approximately **9.5** units.
- (b) The midpoint of \( AB \) is **(-1.5, 0.5)**.

Would you like further details or have any questions?

#### Related Questions:
1. What is the slope of the line segment \( AB \)?
2. How do you derive the distance formula from the Pythagorean theorem?
3. Can you apply the midpoint formula to three-dimensional points?
4. What are the coordinates of point \( A \) reflected over the y-axis?
5. How does changing the coordinates of points affect the midpoint?

#### Tip:
The distance and midpoint formulas are extensions of basic geometry concepts like the Pythagorean theorem and averages, so visualizing these can help when solving similar problems.

For the graph shown to the right, find (a) AB to the nearest tenth and (b) the coordinates of the midpoint of AB.

Learn how to find the distance and midpoint of line segment AB on a coordinate plane. This problem includes using the distance formula and midpoint formula to solve geometric problems. Perfect for students in Grades 8-10.

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