Problem 16 - Algebra for College Students

Question

Verify that \((3,1)\) is the midpoint of the line segment joining \((-2,6)\) and \((8,-4)\).

Step-by-Step Solution

Answer

Yes, \((3,1)\) is the midpoint.

Step 1: Understand the Midpoint Formula

The midpoint of a line segment connecting two points \((x_1, y_1)\) and \((x_2, y_2)\) in the coordinate plane is given by the formula: \( \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \). This formula calculates the average of the x-coordinates and the average of the y-coordinates.

Step 2: Identify the Coordinates of Given Points

The points given are \((-2, 6)\) as \((x_1, y_1)\) and \((8, -4)\) as \((x_2, y_2)\). We'll substitute these coordinates into the midpoint formula to find the midpoint.

Step 3: Substitute the Coordinates into the Midpoint Formula

Substitute \((-2, 6)\) and \((8, -4)\) into the formula:- Calculate the x-coordinate: \( \frac{-2 + 8}{2} = \frac{6}{2} = 3 \).- Calculate the y-coordinate: \( \frac{6 + (-4)}{2} = \frac{2}{2} = 1 \).Thus, the calculated midpoint is \((3, 1)\).

Step 4: Compare the Calculated Midpoint with the Given Midpoint

The calculated midpoint \((3,1)\) matches the given midpoint in the problem. Hence, we have verified that \((3,1)\) is indeed the midpoint of the line segment joining \((-2,6)\) and \((8,-4)\).

Key Concepts

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, is a branch of mathematics that uses a coordinate system to precisely describe locations on a plane. It bridges algebra and geometry through graphing, horizontal and vertical planes, using points plotted as coordinates. These coordinates are defined in pairs, typically denoted as \((x, y)\), where \(x\) is the position on the horizontal axis and \(y\) on the vertical axis.
With coordinate geometry, you can determine the distance between two points, the slope of a line, or, as in this task, the midpoint of a line segment. Understanding how to compute the midpoint is a useful skill that provides insights into the nature of shapes and lines in a plane.
This topic forms a foundation for more advanced topics like calculus, which extends these principles to three dimensions.

Line Segment

A line segment in geometry is a part of a line bounded by two distinct end points. Unlike a line that extends indefinitely in both directions, a line segment has a defined start and end point. You can think of it as a 'piece' of a line.
In our exercise, the line segment connects the coordinates \((-2,6)\) and \((8,-4)\). The importance of the line segment lies in its ability to provide structure within coordinate geometry, allowing us to calculate other properties, such as length, slope, and midpoint.
The midpoint here represents the point that lies exactly halfway between the two endpoints of the segment. Using the midpoint formula helps in confirming whether a given point is truly the midpoint.

Average of Coordinates

The average of coordinates is a fundamental concept used in finding the midpoint of a line segment. It involves calculating the mean or average value of two numbers—in this case, the x-coordinates and y-coordinates of the endpoints of a segment.
To find the midpoint of a segment connecting \((x_1,y_1)\) to \((x_2,y_2)\), you

• Add the x-coordinates: \(x_1 + x_2\).
• Then divide by 2 to find the average: \(\frac{x_1 + x_2}{2}\).
• Repeat the same for the y-coordinates: \(y_1 + y_2\).
• Again divide by 2: \(\frac{y_1 + y_2}{2}\).

The resulting pair \(\left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)\) represents the midpoint. This straightforward yet essential concept helps to balance values and find central positions on a coordinate plane.