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)

Verified step by step guidance

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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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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.

Midpoint Formula

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.

Coordinate Geometry Basics

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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