To solve the given problem, we will use the formulas for the midpoint and the distance between two points:

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(a) Midpoint Formula

The formula to find the midpoint of a line segment joining two points $(x_1, y_1)$ and $(x_2, y_2)$ is:
$\text{Midpoint} = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)$

Substitute $P(-8, 8)$ and $Q(5, -7)$:
$x = \frac{-8 + 5}{2} = \frac{-3}{2}, \quad y = \frac{8 + (-7)}{2} = \frac{1}{2}$

So, the midpoint is: $(x, y) = \left( -\frac{3}{2}, \frac{1}{2} \right)$

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(b) Distance Formula

The formula to find the length of a line segment joining two points $(x_1, y_1)$ and $(x_2, y_2)$ is:
$\text{Length} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$

Substitute $P(-8, 8)$ and $Q(5, -7)$:
$\text{Length} = \sqrt{(5 - (-8))^2 + ((-7) - 8)^2}$ $= \sqrt{(5 + 8)^2 + (-7 - 8)^2}$ $= \sqrt{13^2 + (-15)^2}$ $= \sqrt{169 + 225} = \sqrt{394}$

Thus, the exact length is: $\text{Length} = \sqrt{394}$

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Final Answers:

(a) Midpoint:
$(x, y) = \left( -\frac{3}{2}, \frac{1}{2} \right)$

(b) Length:
$\text{Length} = \sqrt{394}$

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Let me know if you'd like additional details or have any questions!

Expanding Questions:

1. What is the formula for the slope of a line between two points?
2. How would the midpoint formula change in three-dimensional space?
3. Can you express $\sqrt{394}$ in terms of its prime factorization?
4. How do you find the equation of the line passing through points $P$ and $Q$?
5. What is the importance of exact answers in mathematics versus decimal approximations?

Tip:

When working with radicals, always simplify them by breaking them down into prime factors for exact solutions.