For the points P and Q, find (a) the distance d(P, Q) and (b) the coordinates of the midpoint M of line segment PQ. See Examples 1 and 2.
P(-6, -5), Q(6, 10)

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Identify the coordinates of points P and Q: P(-6, -5) and Q(6, 10).

To find the distance d(P, Q), use the distance formula: \(d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\), where \((x_1, y_1)\) and \((x_2, y_2)\) are the coordinates of points P and Q respectively.

Substitute the coordinates into the distance formula: \(d = \sqrt{(6 - (-6))^2 + (10 - (-5))^2}\).

To find the midpoint M of the line segment PQ, use the midpoint formula: \(M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)\).

Substitute the coordinates into the midpoint formula: \(M = \left( \frac{-6 + 6}{2}, \frac{-5 + 10}{2} \right)\).

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Distance Formula

The distance formula calculates the length between two points in a coordinate plane. It is derived from the Pythagorean theorem and given by d = √[(x2 - x1)² + (y2 - y1)²], where (x1, y1) and (x2, y2) are the coordinates of the points.

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The midpoint formula finds the point exactly halfway between two points in a plane. It is calculated as M = ((x1 + x2)/2, (y1 + y2)/2), averaging the x-coordinates and y-coordinates of the endpoints.

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Coordinate geometry connects algebra and geometry using the coordinate plane. Understanding how points, lines, and distances relate through coordinates is essential for solving problems involving distances and midpoints.

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