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. See Examples 2 and 5(a). 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 d(P, Q), use the distance formula: \( d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \).
3. Substitute the coordinates into the distance formula: \( d = \sqrt{(4 - 6)^2 + (6 - (-2))^2} \).
4. Simplify the expression inside the square root to find the distance.
5. To find the midpoint M, use the midpoint formula: \( M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \).

Key Concepts

Distance Formula

The distance formula is used to calculate the distance between two points in a Cartesian plane. It is derived from the Pythagorean theorem and is expressed as d(P, Q) = √((x2 - x1)² + (y2 - y1)²), where (x1, y1) and (x2, y2) are the coordinates of points P and Q, respectively. This formula allows for the determination of the straight-line distance between the two points.

Midpoint Formula

The midpoint formula is used to find the coordinates of the midpoint M of a line segment connecting two points P and Q. It is given by M = ((x1 + x2)/2, (y1 + y2)/2), where (x1, y1) and (x2, y2) are the coordinates of points P and Q. This formula provides the average of the x-coordinates and the y-coordinates of the two points, resulting in the midpoint's coordinates.

Coordinate System

A coordinate system is a two-dimensional plane defined by a horizontal axis (x-axis) and a vertical axis (y-axis). Each point in this system is represented by an ordered pair (x, y), indicating its position relative to the axes. Understanding the coordinate system is essential for applying the distance and midpoint formulas, as it allows for the visualization and calculation of geometric relationships between points.