Problem Breakdown:

We are tasked with finding:

1. The midpoint of the line segment with endpoints $A(-6, 4)$ and $C(4, -6)$.
2. The length of the segment $AC$.

Solution:

a. Midpoint Formula:

The midpoint formula for a line segment with endpoints $(x_1, y_1)$ and $(x_2, y_2)$ is: $M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)$

For $A(-6, 4)$ and $C(4, -6)$: $M_x = \frac{-6 + 4}{2} = \frac{-2}{2} = -1$ $M_y = \frac{4 + (-6)}{2} = \frac{-2}{2} = -1$

Thus, the midpoint is: $M = (-1, -1)$

b. Length of Segment $AC$:

The formula for the length of a line segment is the distance formula: $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$

For $A(-6, 4)$ and $C(4, -6)$: $d = \sqrt{(4 - (-6))^2 + (-6 - 4)^2}$ $d = \sqrt{(4 + 6)^2 + (-6 - 4)^2}$ $d = \sqrt{10^2 + (-10)^2}$ $d = \sqrt{100 + 100} = \sqrt{200}$ $d = 10\sqrt{2}$

Thus, the length of $AC$ is: $10\sqrt{2} \approx 14.14$

Final Answer:

a. Midpoint: $(-1, -1)$
b. Length: $10\sqrt{2}$ or approximately $14.14$

Do you have any questions or need further clarification?

Here are 5 related questions to explore further:

1. How would you find the slope of the line segment $AC$?
2. Can you generalize the midpoint and distance formulas for 3D coordinates?
3. What is the equation of the line passing through points $A(-6, 4)$ and $C(4, -6)$?
4. How would the midpoint change if one of the coordinates were shifted by 2 units?
5. What real-life applications require the use of midpoint or distance formulas?

Tip: Always double-check calculations, especially when working with square roots or negative numbers!