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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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

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.

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)