91Ó°ÊÓ

Coordinates are essential when dealing with the geometry of points on the plane. They describe the exact spike on a graph where the point is located. For a two-dimensional plane, we use an ordered pair

• The first number is the x-coordinate, which shows the horizontal position.
• The second number is the y-coordinate, which shows the vertical position.

In our problem, the endpoints of the line segment are given as \((\sqrt{50}, -6)\) and \((\sqrt{2}, 6)\). These point coordinates tell us where each endpoint lies in the plane. Understanding coordinates helps us easily perform calculations, such as finding a midpoint, by providing a clear numerical framework.

A line segment is a part of a line that is bounded by two endpoints. It is one-dimensional and always has a finite length. Unlike a line that extends indefinitely in both directions, a line segment is just the slice between two specific points. In the given problem, the endpoints are \((\sqrt{50}, -6)\) and \((\sqrt{2}, 6)\). Thus, the line segment joins these two points. Understanding the concept of a line segment is crucial because it allows us to apply formulas like the midpoint formula. Knowing the endpoints, we can compute the distance, the midpoint, and other geometric properties relevant to the segment.

The square root is a mathematical operation that finds a number that, when multiplied by itself, results in the original number. For example, the square root of 4 is 2, because \(2 \times 2 = 4\). In our context, we have \(\sqrt{50}\) and \(\sqrt{2}\) as part of the coordinates for the endpoints of the line segment. Having square roots in coordinate calculations might seem intimidating, but it's quite manageable. When calculating midpoints, you can sum the square root expressions directly, as they are just numbers, and then divide by two to find the coordinate of the midpoint. Be sure to approximate if you need a decimal form, but exact calculations work perfectly for midpoint determination.

The midpoint of a line segment is simply the average point, or the center, of the line segment. It is calculated by taking the average of the x-coordinates and the y-coordinates separately. Here's the formula for finding the midpoint \((M)\) of a segment with endpoints \((x_1, y_1)\) and \((x_2, y_2)\):

• Midpoint's x-coordinate: \( \frac{x_1 + x_2}{2} \)
• Midpoint's y-coordinate: \( \frac{y_1 + y_2}{2} \)

For our endpoints \((\sqrt{50}, -6)\) and \((\sqrt{2}, 6)\), the midpoint's x-coordinate is \( \frac{\sqrt{50} + \sqrt{2}}{2} \), and the y-coordinate is calculated as \( \frac{-6 + 6}{2} = 0 \). Thus, the midpoint coordinate is \( \left( \frac{\sqrt{50} + \sqrt{2}}{2}, 0 \right) \). Understanding how to find the midpoint is important because it offers a way to determine balance and symmetry in geometric figures.