Question

The coordinates of the endpoints of a segment are given. Find the coordinates of the midpoint of each segment. \((a, b),(0,0)\)

Short answer

The midpoint is \( \left( \frac{a}{2}, \frac{b}{2} \right) \).

Step-by-step solution

Recall the Midpoint Formula

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

Identify the Endpoints

Identify the coordinates of the endpoints from the problem: \((x_1, y_1) = (a, b)\) and \((x_2, y_2) = (0, 0)\).

Apply the Midpoint Formula

Substitute the coordinates of the endpoints into the midpoint formula: \( \left( \frac{a + 0}{2}, \frac{b + 0}{2} \right) \).

Simplify the Expression

Simplify the expressions inside the formula by doing the arithmetic: \( \left( \frac{a}{2}, \frac{b}{2} \right) \). These are the coordinates of the midpoint.

Key concepts

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, is a branch of geometry where the positions of points are defined using coordinates. These coordinates are typically in two-dimensional space, using a pair of numbers to define the exact location on a plane. Understanding coordinates is foundational for locating points, lines, and shapes within a plane.

This system uses two perpendicular lines called axes, the x-axis, and the y-axis. The point where these axes intersect is known as the origin, labeled as (0, 0) .

• The x-coordinate signifies horizontal distance from the origin,
• and the y-coordinate signifies vertical distance from the origin.

This approach allows us to place both geometry and algebra in the same frame, making it easier to solve geometric problems.

Applied problems like finding a midpoint use these concepts to determine the center between two points in a plane. Without coordinate geometry, problems that involve specific locations in a plane would be impossible to solve efficiently.

Endpoints

Endpoints are the two distinct points at the ends of a line segment. In coordinate geometry, they are often defined using ordered pairs (x_1, y_1) and (x_2, y_2) . Understanding endpoints is crucial because many geometric properties and calculations, such as length, slope, and midpoints, are based on the location of these points.

Each endpoint provides a definitive position on a coordinate plane, allowing us to calculate specific attributes of the segment. For example, when we speak of endpoints in the exercise, one point is (a, b) and another is (0, 0) . These coordinates serve as the fundamental building blocks for more complicated calculations, such as determining the midpoint or distance between the two endpoints.

• They allow for geometric construction and calculation,
• and are essential in defining the properties of the segment they form.

In essence, understanding the location and role of endpoints simplifies many computations in coordinate geometry.

Midpoint Calculation

The midpoint calculation is a simple yet powerful tool in coordinate geometry that finds the center point of a line segment. The midpoint represents an average of the x-coordinates and y-coordinates of the segment's endpoints. To calculate the midpoint, we use the formula: \[\left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)\] This formula effectively finds the midway point between two coordinates by averaging them. Let's say our endpoints are (a, b) and (0, 0); substituting these into the formula gives us the midpoint as \[\left( \frac{a}{2}, \frac{b}{2} \right)\]

• The process is straightforward: Add the two x-coordinates and divide by 2 to find the midpoint's x-coordinate.
• Similarly, add the y-coordinates and divide by 2 to find the midpoint's y-coordinate.

This midpoint is significant because it not only splits the segment into two equal parts but can also be a critical element in other mathematical operations and geometric interpretations.