Problem 56 - Elementary and Intermediate Algebra

Question

Find the midpoint of the line segment with the given endpoints. \((10,12),(0,0)\)

Step-by-Step Solution

Answer

The midpoint is (5, 6).

Step 1: Identify the Formula

To find the midpoint of a line segment, we use the formula \( \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \). This formula calculates the average of the x-coordinates and the y-coordinates.

Step 2: Extract Endpoint Coordinates

Identify the coordinates of the endpoints. Here, the endpoints are \((x_1, y_1) = (10, 12)\) and \((x_2, y_2) = (0, 0)\).

Step 3: Substitute into the Formula

Substitute the endpoint coordinates into the midpoint formula:\[ \left( \frac{10 + 0}{2}, \frac{12 + 0}{2} \right) \].

Step 4: Calculate Each Coordinate

Calculate the x and y coordinates of the midpoint separately:For x: \( \frac{10 + 0}{2} = \frac{10}{2} = 5 \),For y: \( \frac{12 + 0}{2} = \frac{12}{2} = 6 \).

Step 5: Write the Midpoint

Combine the calculated coordinates to determine the midpoint:\((5, 6)\).

Key Concepts

Coordinate Geometry

Coordinate Geometry is a branch of mathematics that utilizes algebra and geometry to explore the relationships between different points on a plane. In this context, each point is defined by a pair of numerical coordinates. These coordinates are usually written as

• (x, y) - where x represents the horizontal position, and y represents the vertical position.

Coordinate Geometry allows us to perform various operations such as finding distances between points, locating midpoints, and defining geometric shapes and figures on a coordinate plane. The grid-like structure of the coordinate plane makes it easier to handle these operations through mathematical formulas. Understanding these basics enables students to dive into more complex topics like circles, ellipses, and hyperbolas. This makes Coordinate Geometry a fundamental part of algebra and geometry studies.

Line Segment

A line segment is a straight connection between two points on a coordinate plane. These two points are called the endpoints of the segment. Unlike a line, which extends infinitely in both directions, a line segment has a finite length. This distinct characteristic helps in:

• Measuring distances
• Calculating midpoints
• Analyzing the relative positions of the points

In the given exercise, the line segment connects the endpoints

• (10, 12) and (0, 0)

Understanding what a line segment represents is essential for applying formulas in Coordinate Geometry effectively. Once you grasp how to identify and interpret line segments, using algebraic formulas to solve geometric problems becomes much more intuitive.

Algebraic Formula

An algebraic formula serves as a mathematical tool that helps you compute unknown values when certain conditions are met. In the context of our exercise, we use the Midpoint Formula:

• \( \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \)

This formula allows us to determine the midpoint of a line segment by averaging the coordinates of its endpoints. By substituting the values from the exercise endpoints

• (10,12) and (0,0)

into the formula, the midpoint is calculated as (5, 6). This straightforward formula makes use of basic algebraic operations like addition and division to achieve a valuable geometric result. Algebraic formulas like this underpin much of Coordinate Geometry, streamlining computations and helping students focus on understanding deeper concepts.