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(8, 2), Q(3, 5)

검증된 단계별 안내

Identify the coordinates of points P and Q. Here, P has coordinates (8, 2) and Q has coordinates (3, 5).

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

Substitute the coordinates into the distance formula: \(d(P, Q) = \sqrt{(3 - 8)^2 + (5 - 2)^2}\).

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 of P and Q into the midpoint formula: \(M = \left( \frac{8 + 3}{2}, \frac{2 + 5}{2} \right)\).

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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 plane using their coordinates. It is derived from the Pythagorean theorem and given by d = √[(x2 - x1)² + (y2 - y1)²]. This formula helps find the straight-line distance between points P and Q.

Midpoint Formula

The midpoint formula finds the point exactly halfway between two given points in a coordinate plane. It is calculated by averaging the x-coordinates and y-coordinates separately: M = ((x1 + x2)/2, (y1 + y2)/2). This gives the coordinates of the midpoint M of segment PQ.

Coordinate Geometry Basics

Coordinate geometry connects algebra and geometry by representing geometric figures using coordinates. Understanding how points, lines, and distances are expressed in the coordinate plane is essential for applying formulas like distance and midpoint effectively.

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(8, 2), Q(3, 5)