Problem 30 - College Algebra

Question

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

Step-by-Step Solution

Answer

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

Step 1: Identify the Given Points

First, identify the given points. The two 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

Next, apply the midpoint formula. The midpoint M of a line segment with endpoints (x1, y1) and (x2, y2) is given by \((\frac{{x1+x2}}{2}, \frac{{y1+y2}}{2})\). Substituting the values, we get M = \((\frac{{\sqrt{50}+\sqrt{2}}}{2}, \frac{{-6+6}}{2})\).

Step 3: Simplify the Expression

Simplify the expression to get the coordinates of the midpoint. This results in M = \((\frac{{\sqrt{50}+\sqrt{2}}}{2}, 0)\).

Key Concepts

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, is a branch of mathematics that involves the study of geometric figures using a coordinate system. This method allows for the representation of points in a plane using ordered pairs of numbers, typically referred to as the x (horizontal) and y (vertical) coordinates.

For example, the endpoint of a line segment can be expressed as a pair \( (x, y) \), where \( x \) corresponds to the position along the horizontal axis, and \( y \) signifies the position on the vertical axis. Understanding coordinate geometry is essential in solving problems related to the distance between points, the slopes of lines, and finding the midpoint of a line segment, as in the given exercise.

Line Segment Endpoints

The endpoints of a line segment are the points where the segment terminates. These points can be visualized on a graph as the 'corners' of the line segment. In coordinate geometry, the endpoints are expressed using their coordinates. Take the given problem, which provides us with two endpoints, \( (\sqrt{50}, -6) \) and \( (\sqrt{2}, 6) \). Each of these coordinates includes an \( x- \) and \( y- \)component, which designate the endpoint's location on the coordinate plane.

When working with line segment endpoints, one often needs to calculate important values like the length of the segment, its midpoint, or determine if it's perpendicular or parallel to another segment. The midpoint, as calculated in our exercise, precisely represents the halfway point between the two endpoints of the line segment.

Algebraic Expressions

An algebraic expression is a mathematical phrase that can include numbers, variables, and operation symbols. These expressions are used to represent values in a general form and can be as simple as a single term or as complex as a multi-term expression. In the exercise, the midpoint formula involves an algebraic expression that combines the coordinates of the endpoints to calculate the midpoint.

The process to find this midpoint involves adding the x-coordinates of the endpoints, dividing by two for the x-value of the midpoint, then doing the same for the y-coordinates to get the y-value. Here, the algebraic expression for the midpoint \( M \) is \( (\frac{\sqrt{50} + \sqrt{2}}{2}, \frac{-6 + 6}{2}) \), which simplifies to \( (\frac{\sqrt{50} + \sqrt{2}}{2}, 0) \). Algebraic expressions are a cornerstone in the study of mathematics as they form the basis for complicated equations and formulae used across various areas of math.