Problem 87 - Intermediate Algebra

Question

Find the midpoint of the line segment whose endpoints are given. See Example 7 $$ (6,-8),(2,4) $$

Step-by-Step Solution

Answer

The midpoint is (4, -2).

Step 1: Understand the Formula

The formula to find the midpoint \( M \) of a line segment with endpoints \((x_1, y_1)\) and \((x_2, y_2)\) is \( M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \).

Step 2: Identify the Endpoints

Here, the given endpoints of the line segment are \((6, -8)\) and \((2, 4)\). We identify \(x_1 = 6\), \(y_1 = -8\), \(x_2 = 2\), and \(y_2 = 4\).

Step 3: Apply the Formula to X-coordinates

To find the x-coordinate of the midpoint, apply the formula: \( \frac{x_1 + x_2}{2} = \frac{6 + 2}{2} = \frac{8}{2} = 4 \).

Step 4: Apply the Formula to Y-coordinates

To find the y-coordinate of the midpoint, apply the formula: \( \frac{y_1 + y_2}{2} = \frac{-8 + 4}{2} = \frac{-4}{2} = -2 \).

Step 5: Combine the Coordinates

The midpoint of the line segment with endpoints \((6, -8)\) and \((2, 4)\) is \((4, -2)\).

Key Concepts

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, is a branch of mathematics where we use coordinates to help us understand geometric figures like lines, circles, and polygons. It combines algebra and geometry to offer a powerful toolbox for solving geometric problems.

Whenever we talk about coordinate geometry, we're usually referring to graphs plotted on a Cartesian plane. In this plane, each point is labeled with a pair of numbers called coordinates, written as \(x, y\). This allows us to move from abstract cars and shapes to precise mathematical work.

Understanding the layout of coordinate geometry helps with solving problems such as finding distances between points, the equation of lines, and much more. It forms the foundation for many concepts in higher mathematics and physics.

Midpoint Formula

The midpoint of a line segment is the point that divides the segment into two equal parts. The midpoint formula is essential for finding this point precisely. If we know the endpoints \(x_1, y_1\) and \(x_2, y_2\) of a line segment, we can find the midpoint by using the formula:
\[ M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \]
This formula represents the average of the x-coordinates and the y-coordinates of the endpoints.

Using this method is relatively simple:

• First, add the two x-coordinates together and divide by 2 to find the x-coordinate of the midpoint.
• Second, add the two y-coordinates together and divide by 2 to find the y-coordinate of the midpoint.
• Finally, combine these two averages to form the complete set of coordinates for the midpoint.

It's a straightforward process that provides a clear mathematical way to find a central point on a line segment.

Line Segments

A line segment is a part of a line that has two endpoints. Unlike a line, which extends infinitely in both directions, a line segment has a definite length and does not continue. This makes it easier to work with when calculating things like midpoints or lengths.

Understanding line segments is crucial to mastering coordinate geometry, as they are used to represent simple connections between points on a plane. And when using the midpoint formula, what we're really doing is finding the center of these connections.

Line segments are found all over geometry problems, such as in polygons where each side is a line segment, or when calculating the bisector of an angle. Knowing how to work with line segments is key to unlocking more intricate geometric properties.

• When working with line segments, it's important to measure not just length but also the location of the segment on a coordinate plane, using endpoints.
• Understanding the segment’s position helps with tasks such as finding midpoints, making complex shapes easier to analyze.

This understanding makes the calculation of various other geometric elements, like angles and distances, more straightforward.