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.