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 two points: \(P(3, -1)\) and \(Q(-4, 5)\).
2. Use the distance formula to find the distance between points \(P\) and \(Q\): \[\text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\] Substitute \(x_1 = 3\), \(y_1 = -1\), \(x_2 = -4\), and \(y_2 = 5\) into the formula.
3. Calculate the differences in the coordinates: \[x_2 - x_1 = -4 - 3\] \[y_2 - y_1 = 5 - (-1)\]
4. Square each difference, add them, and then take the square root to find the distance: \[\sqrt{(-4 - 3)^2 + (5 - (-1))^2}\]
5. To find the midpoint coordinates, use the midpoint formula: \[\left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)\] Substitute the values of \(x_1\), \(x_2\), \(y_1\), and \(y_2\) to get the midpoint.

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 the square root of the sum of the squares of the differences in x-coordinates and y-coordinates: distance = √[(x2 - x1)² + (y2 - y1)²].

Midpoint Formula

The midpoint formula finds the point exactly halfway between two given points in the coordinate plane. It is calculated by averaging the x-coordinates and the y-coordinates separately: midpoint = ((x1 + x2)/2, (y1 + y2)/2). This point divides the segment into two equal parts.

Coordinate Plane and Points

Understanding the coordinate plane involves recognizing that each point is represented by an ordered pair (x, y), where x is the horizontal position and y is the vertical position. This framework allows for geometric interpretations and calculations such as distance and midpoint between points.