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)

Verified step by step guidance

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: \(M = \left( \frac{8 + 3}{2}, \frac{2 + 5}{2} \right)\).

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.

Video duration:

Was this helpful?

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 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.

Recommended video: