Problem 93 - Algebra A Combined Function

Question

Find the midpoint of each line segment whose endpoints are given. \(\left(\frac{1}{2}, \frac{3}{8}\right) ;\left(-\frac{3}{2}, \frac{5}{8}\right)\)

Step-by-Step Solution

Answer

The midpoint of the line segment is \((-\frac{1}{2}, \frac{1}{2})\).

Step 1: Identify the Endpoint Coordinates

The problem provides the endpoints of the line segment as \( \left(\frac{1}{2}, \frac{3}{8}\right) \) and \( \left(-\frac{3}{2}, \frac{5}{8}\right) \). Here, the first point \( A \) has coordinates \( (x_1, y_1) = \left(\frac{1}{2}, \frac{3}{8}\right) \) and the second point \( B \) has coordinates \( (x_2, y_2) = \left(-\frac{3}{2}, \frac{5}{8}\right) \).

Step 2: Apply the Midpoint Formula

The formula to find the midpoint \( M \) of a line segment is given by \( M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \). Let's apply this formula to the endpoint coordinates.

Step 3: Calculate the Midpoint's X-Coordinate

Using the formula \( x_M = \frac{x_1 + x_2}{2} \), substitute the values for \( x_1 \) and \( x_2 \): \[ x_M = \frac{\frac{1}{2} + (-\frac{3}{2})}{2} = \frac{\frac{1}{2} - \frac{3}{2}}{2} = \frac{-1}{2} \]

Step 4: Calculate the Midpoint's Y-Coordinate

Using the formula \( y_M = \frac{y_1 + y_2}{2} \), substitute the values for \( y_1 \) and \( y_2 \): \[ y_M = \frac{\frac{3}{8} + \frac{5}{8}}{2} = \frac{\frac{8}{8}}{2} = \frac{1}{2} \]

Step 5: Write Down the Midpoint Coordinates

Combine the results from Steps 3 and 4 to get the midpoint of the line segment. Thus, the midpoint is \( M = \left( -\frac{1}{2}, \frac{1}{2} \right) \).

Key Concepts

Coordinate Geometry

Coordinate geometry, sometimes called analytic geometry, combines algebra and geometry to study the position of points, lines, and figures on a plane. It uses a coordinate system, typically the Cartesian coordinate system, which comprises two perpendicular axes: the x-axis and the y-axis. Together, these axes create a grid where any point can be represented with a pair of numerical coordinates. In coordinate geometry, every point has a unique address made up of an x-coordinate (horizontal position) and a y-coordinate (vertical position). The Cartesian coordinate system makes it easy for us to describe geometric shapes and figure out exactly where things are placed.

• The x-axis is horizontal, and the y-axis is vertical.
• Coordinates are written in the form (x, y).
• Each coordinate tells you how far to go along the x-axis and the y-axis to reach the point.

In the context of finding a midpoint, coordinate geometry provides a framework where we can use numerical methods to determine points that are exactly halfway along a line segment on this grid.

Line Segment

A line segment is one of the building blocks of geometry. It's a portion of a line bounded by two distinct endpoint coordinates. These endpoints give the line segment a fixed length, unlike an infinite line that stretches endlessly in both directions. In our exercise, we're particularly interested in understanding the midpoint of this line segment.

• A line segment is defined by two endpoints.
• It has a definite length, unlike a line.
• The distance between the two endpoints marks the extent of the line segment.

When working with line segments in coordinate geometry, you can calculate various things about them, like their length or midpoint, by relying on the coordinates of the endpoints. Understanding line segments can be especially helpful when you're comparing different parts of some shape or structure within a plane.

Fractions

Fractions are a way to represent parts of a whole or a division of quantities. They are essential when dealing with coordinates that aren't whole numbers, especially in coordinate geometry where precision is necessary. In our specific task of finding a midpoint, fractions will be crucial as they allow us to perform precise arithmetic without converting entirely to decimal form.

• A fraction consists of a numerator (top number) and a denominator (bottom number).
• The numerator represents how many parts are being considered, while the denominator tells into how many equal parts the whole is divided.
• Fractions can be added, subtracted, multiplied, and divided to find precise outcomes in calculations.

When calculating the midpoint, knowing how to manage fractions allows us to properly handle the addition and division of the endpoint coordinates, ensuring accuracy in finding that middle point on our line segment.

Endpoint Coordinates

Endpoint coordinates are the specific points that mark the beginning and end of a line segment. In coordinate geometry, these points are usually represented as ordered pairs (x, y), which denote their position on the Cartesian plane. For any given line segment, the coordinates of the endpoints allow us to perform calculations, like finding a midpoint, using specific geometric formulas.

• The endpoints define the boundaries of the line segment.
• Each endpoint is represented by a pair of coordinates (x, y).
• The position of these points influences various geometric properties of the segment.

To find the midpoint of a line segment, we position the coordinates of its endpoints into the midpoint formula. This formula effectively averages out the positions of the x-coordinates and y-coordinates, resulting in a new coordinate—a point that lies exactly between the two endpoints on the segment.