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

Verified step by step guidance

1. To find the distance between two points \( P(x_1, y_1) \) and \( Q(x_2, y_2) \), use the distance formula: \( d(P, Q) = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \).
2. Substitute the given coordinates of points \( P(8, 2) \) and \( Q(3, 5) \) into the distance formula: \( d(P, Q) = \sqrt{(3 - 8)^2 + (5 - 2)^2} \).
3. Simplify the expression inside the square root: \( d(P, Q) = \sqrt{(-5)^2 + 3^2} \).
4. Calculate the squares and sum them: \( d(P, Q) = \sqrt{25 + 9} \).
5. 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: \( M = \left( \frac{8 + 3}{2}, \frac{2 + 5}{2} \right) \).

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 us to find 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 accurate identification and manipulation of points in the plane.