For the points P and Q, find (a) the distance d(P, Q) and (b) the coordinates of the mid-point M of line segment PQ. P(6,-2), Q(4,6)

Verified step by step guidance

1. Identify the coordinates of points P and Q: P(6, -2) and Q(4, 6).
2. 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.
3. Substitute the coordinates into the distance formula: \(d(P, Q) = \sqrt{(4 - 6)^2 + (6 - (-2))^2}\).
4. To find the midpoint M of the line segment PQ, use the midpoint formula: \(M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)\).
5. Substitute the coordinates into the midpoint formula: \(M = \left( \frac{6 + 4}{2}, \frac{-2 + 6}{2} \right)\).

Key Concepts

Distance Formula

The distance formula calculates the length between two points in the 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 given points. It is calculated by averaging the x-coordinates and the y-coordinates separately: M = ((x1 + x2)/2, (y1 + y2)/2). This gives the coordinates of the midpoint.

Coordinate Plane and Points

Understanding the coordinate plane involves knowing how points are represented as ordered pairs (x, y). Each point's position is determined by its horizontal (x) and vertical (y) distances from the origin, which is essential for applying distance and midpoint formulas.