Problem
For the point $P(-11,-16)$ and $Q(-4,-11)$, find the distance $d(P, Q)$ and the coordinates of the midpoint $M$ of the segment $P Q$. What is the distance? (Simplify your answer. Type an exact answer, using radicals as need.) What are the coordinates of the midpoint M? (Simplify your answer. Type an ordered pair, using integers or fractions.)
Solution
Step 1 ：Given points P(-11, -16) and Q(-4, -11).
Step 2 ：Use the distance formula: $d(P, Q) = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$.
Step 3 ：Plug in the coordinates: $d(P, Q) = \sqrt{(-4 - (-11))^2 + (-11 - (-16))^2} = \sqrt{49 + 25} = \sqrt{74}$.
Step 4 ：Use the midpoint formula: $M = (\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2})$.
Step 5 ：Plug in the coordinates: $M = (\frac{-11 + (-4)}{2}, \frac{-16 + (-11)}{2}) = (-7.5, -13.5)$.
Step 6 ：Final Answer: The distance between points P and Q is \(\boxed{\sqrt{74}}\) and the coordinates of the midpoint M are \(\boxed{(-7.5, -13.5)}\).
