Problem 55 - Algebra 2

Question

Find the midpoint of the line segment with endpoints having the given coordinates. $$ (2,-9),(-4,5) $$

Step-by-Step Solution

Answer

The midpoint is (-1, -2).

Step 1: Understand the midpoint formula

The formula to find the midpoint of a line segment with endpoints \((x_1, y_1)\) and \((x_2, y_2)\) is given by: \( \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \). This formula finds the average of the x-coordinates and the y-coordinates of the endpoints.

Step 2: Identify the coordinates of the endpoints

You are given two endpoints: \((2, -9)\) and \((-4, 5)\). Here, \(x_1 = 2\), \(y_1 = -9\), \(x_2 = -4\), and \(y_2 = 5\).

Step 3: Calculate the average of the x-coordinates

Using the x-coordinates \(x_1 = 2\) and \(x_2 = -4\), compute the average: \( \frac{2 + (-4)}{2} = \frac{2 - 4}{2} = \frac{-2}{2} = -1\).

Step 4: Calculate the average of the y-coordinates

Using the y-coordinates \(y_1 = -9\) and \(y_2 = 5\), compute the average: \( \frac{-9 + 5}{2} = \frac{-4}{2} = -2\).

Step 5: Form the coordinates of the midpoint

Combining both averages, the midpoint is \((-1, -2)\).

Key Concepts

Line Segment

A line segment is an essential concept in geometry. It is a part of a line that is bounded by two distinct endpoints. Line segments have a definite length and they can be measured in units such as centimeters or inches. Unlike lines, line segments do not extend infinitely in any direction.
To visualize it in simple terms, imagine a pencil resting flat on a desk; the pencil represents the line segment, and the two sharpened ends serve as the endpoints. Each segment connects one endpoint to the other in a straight path. In our original exercise, the endpoints are given as the coordinates (2, -9) and (-4, 5).
This means, between these two points, there is a specific section of a straight line, which we are working to find the midpoint along.

Coordinates

Coordinates help us locate points in space. In mathematics, especially in geometry, coordinates are expressed as pairs of numbers. Each pair represents a specific point on a grid or plane. The most common type is the Cartesian coordinate system, using the format (x, y).

• The first number in the pair—the x-coordinate—shows horizontal distance from a specified origin (often the center of the plane).
• The second number—the y-coordinate—indicates vertical distance from the origin.

In our problem, the endpoints of the line segment are given as coordinates: (2, -9) and (-4, 5).
Understanding coordinates is crucial as it allows us to perform calculations and solve problems that relate to position, such as finding the midpoint of the segment.

Average

The concept of an average is foundational when calculating the midpoint of a line segment. An average is a value that represents the central or typical value in a set of data. In mathematics, when dealing with coordinates, the term average means the mean value of numbers you are working with.

For the midpoint of a line segment, you need the average of the x-coordinates and the y-coordinates separately. This is because the midpoint lies directly between the two endpoints on each axis:

• To find the average of the x-coordinates, you add the x-values and divide by 2: \( \frac{x_1 + x_2}{2} \).
• The same is applied to the y-coordinates: \( \frac{y_1 + y_2}{2} \).

Taking these averages gives us the coordinates of our desired midpoint. In the given exercise, the averages are calculated to be -1 for x and -2 for y.

Endpoints

Endpoints are specific points that mark the boundaries or limits of a line segment. Each endpoint has its own pair of coordinates that determine its position on a plane. In the context of a line segment, endpoints are crucial because they define the entire segment.
Imagine endpoints like the two halves of a chocolate bar's square pieces; they tell you where the break in the bar is, with the entire chocolate square between them standing as your line segment.

• These endpoints are crucial as they allow you to calculate the length of a line segment.
• Moreover, they help in locating the midpoint by averaging their coordinates.

In our example, the endpoints are given as the coordinates (2, -9) and (-4, 5). Knowing these points allows us to apply the midpoint formula effectively and compute the exact central point of our segment.