Problem 30 - College Algebra

Question

Find the midpoint of each line segment with the given endpoints. $$ (\sqrt{50},-6) \text { and }(\sqrt{2}, 6) $$

Step-by-Step Solution

Answer

The midpoint of the line segment with endpoints (\sqrt{50},-6) and (\sqrt{2}, 6) is (\sqrt{13}, 0).

Step 1: Identify the Coordinates of the Endpoints

We are given two sets of coordinates which are the endpoints in a two-dimensional plane. The endpoints are \((\sqrt{50},-6)\) and \((\sqrt{2}, 6)\). So, \(x1= \sqrt{50}\), \(y1= -6\), \(x2= \sqrt{2}\) and \(y2= 6\).

Step 2: Apply the Midpoint Formula

The midpoint formula is given by \((\frac{{x1+x2}}{2}, \frac{{y1+y2}}{2})\). Substituting the endpoint coordinates from Step 1 into the midpoint formula gives: \(\frac{{\sqrt{50}+\sqrt{2}}}{2}, \frac{{-6+6}}{2}\).

Step 3: Calculate the Midpoint

Evaluate the expressions to find the coordinates of the midpoint. This provides \(\frac{{\sqrt{50}+\sqrt{2}}}{2}= \sqrt{13}\) for x and \(\frac{{-6+6}}{2}= 0\) for y. Therefore, the midpoint of the line segment with the given endpoints is at \((\sqrt{13}, 0)\) .

Key Concepts

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, is a branch of mathematics that uses algebra to study geometric properties and relationships. It's a way to represent geometric figures and analyze their properties by using a coordinate system, like the Cartesian plane.

This system allows you to plot points using pairs of numbers which are called coordinates. Each point is described by a set of coordinates \(x, y\). The horizontal axis is the x-axis, and the vertical axis is the y-axis. By plotting points and drawing lines, you can visually understand the relationship between different geometric objects.

In our exercise, we used coordinate geometry to find the midpoint of a line segment with given endpoints. Understanding the position of these points on a two-dimensional plane helps you to calculate various geometric properties, such as distance, slope, and midpoint.

Line Segments

A line segment is a part of a line that is bounded by two distinct endpoints. Unlike a line, a line segment does not extend indefinitely. In coordinate geometry, we define these endpoints using coordinate pairs \(x_1, y_1\) and \(x_2, y_2\).

To find the midpoint of a line segment, you take the average of the x-coordinates and the y-coordinates of the endpoints. This will give you the coordinates of a point that lies exactly in the middle of the segment.

Line segments are fundamental in geometry as they form the basis of more complex shapes and constructs. In our example, identifying the endpoints \((\sqrt{50},-6)\) and \((\sqrt{2}, 6)\) helped us to use the midpoint formula effectively to find the midpoint.

Distance in Geometry

Distance in geometry refers to the measure of space between two points in a coordinate plane. The Euclidean distance formula is often used, but when dealing with midpoints, knowing how to calculate evenly split line segments is equally important.

The distance between two points \(x_1, y_1\) and \(x_2, y_2\) can be calculated using the distance formula: \(\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\\).

However, when finding midpoints, we focus on splitting the line segment into two equal parts. This involves calculating the average of the x-coordinates and y-coordinates of the endpoints, as shown in our exercise. The midpoint serves as a point equidistant from both endpoints, providing a useful reference in geometric constructions and proofs.