Find the distance between each pair of points, and give the coordinates of the midpoint of the line segment joining them. P(3, -1), Q(-4, 5)

Verified step by step guidance

1. Identify the coordinates of the points: P(3, -1) and Q(-4, 5).
2. Use the distance formula: $d=\sqrt{(x_2-x_1{)}^2+(y_2-y_1{)}^2}$ to find the distance between the points.
3. Substitute the coordinates into the distance formula: $d=\sqrt{((-4)-3{)}^2+(5-(-1){)}^2}$.
4. Simplify the expression inside the square root to find the distance.
5. Use the midpoint formula: $M=(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2})$ to find the midpoint of the line segment.

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 = √((x2 - x1)² + (y2 - y1)²), where (x1, y1) and (x2, y2) are the coordinates of the two points. This formula allows for the determination of the straight-line distance between any two points.

Midpoint Formula

The midpoint formula is used to find the point that is exactly halfway between two given points in a coordinate system. It is calculated using the formula M = ((x1 + x2)/2, (y1 + y2)/2), where (x1, y1) and (x2, y2) are the coordinates of the two points. This concept is essential for determining the center point of a line segment connecting two points.

Coordinate Plane

A coordinate plane is a two-dimensional surface defined by a horizontal axis (x-axis) and a vertical axis (y-axis). Each point on the plane is represented by an ordered pair (x, y), indicating its position relative to the axes. Understanding the coordinate plane is crucial for visualizing points, lines, and shapes, and for applying formulas like the distance and midpoint formulas effectively.