Question

The coordinates of the endpoints of a segment are given. Find the coordinates of the midpoint of each segment. (Lesson 2-5) $$(0,0),(x, y)$$

Short answer

The midpoint is \(\left( \frac{x}{2}, \frac{y}{2} \right)\).

Step-by-step solution

Understand the Problem

You're given two endpoints of a line segment: \((0,0)\) and \((x, y)\). The task is to find the midpoint of this segment.

Recall the Midpoint Formula

The midpoint \((M_x, M_y)\) of a segment with endpoints \((x_1, y_1)\) and \((x_2, y_2)\) is given by the formula: \[(M_x, M_y) = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right).\]

Apply the Midpoint Formula

Substitute the given endpoints \((0, 0)\) and \((x, y)\) into the formula. Let \((x_1, y_1) = (0, 0)\) and \((x_2, y_2) = (x, y)\). The midpoint is calculated as follows: \[\left( \frac{0 + x}{2}, \frac{0 + y}{2} \right) = \left( \frac{x}{2}, \frac{y}{2} \right).\]

State the Midpoint

The coordinates of the midpoint of the segment are \(\left( \frac{x}{2}, \frac{y}{2} \right)\).

Key concepts

Coordinate Geometry

Coordinate Geometry is a branch of mathematics that allows us to use algebra to investigate geometric properties. It involves defining and analyzing geometrical shapes using a coordinate plane, which is essentially a two-dimensional number line. This plane allows us to map out locations using a pair of coordinates (x, y). Each point on this plane is defined by its unique coordinates, which helps in identifying specific positions.

In the exercise we're analyzing, we use coordinate geometry to work with endpoints on a line segment and find the midpoint. This process makes it easier to visualize problems and solutions related to shapes and lines by providing a numerical method to describe points. Thus, understanding coordinate geometry is crucial as it bridges algebra and geometry, allowing us to calculate distances, midpoints, and even slopes with ease.

The coordinate system consists of:

• The x-axis, which runs horizontally
• The y-axis, which runs vertically
• The origin, where both axes meet, represented by the point (0,0)

This system is helpful not only for solving theoretical mathematical problems but also for practical applications like mapping and computer graphics.

Line Segment

A line segment is a part of a line that is bounded by two distinct endpoints. Unlike a line, which extends infinitely in both directions, a line segment has a definite beginning and end. In the realm of coordinate geometry, a line segment is defined by its two endpoints, each with its own set of coordinates.

For instance, in our given exercise, we have a line segment with endpoints at

• (0,0)
• (x, y)

Studying line segments is essential because they form the foundation for more complex shapes like polygons. To determine various properties of a line segment, such as its length or midpoint, you need to work with the coordinates of its endpoints.

The concepts of line segments become even more exciting when we use the midpoint formula. The midpoint is the equidistant point from both endpoints along the segment. The formula allows us to calculate this exact point easily, transforming line segment analysis from a purely theoretical task into practical problems solvable with basic algebraic knowledge.

Endpoints

Endpoints define the boundaries of a line segment in coordinate geometry. They are crucial because they determine the precise location and length of the line segment.

When given endpoints, such as (0,0) and (x, y), they identify the extremities of the segment. In this particular exercise, the points are straightforward, with one being the origin (0, 0), making it simpler to calculate further properties like the midpoint.

With endpoints:

• The x-coordinate indicates how far left or right the point is
• The y-coordinate shows how far up or down the point is

Understanding the fixed positions of endpoints on a plane helps solve geometric problems like finding distances between points or midpoints of segments. This method of using endpoints is critical because it enables precise calculation in constructing geometrical figures and solving real-world problems, where measurement and accuracy are paramount.