Problem 20 - College Algebra Essentials

Question

find the midpoint of each line segment with the given endpoints. $$ (10,4) \text { and }(2,6) $$

Step-by-Step Solution

Answer

The midpoint of the line segment with endpoints at (10,4) and (2,6) is (6,5).

Step 1: Identify the Coordinates

Identify the coordinates of the endpoints. Here, the first point A is represented as \(A(x_1, y_1) = (10, 4)\) and the second point B is represented as \(B(x_2, y_2) = (2, 6)\).

Step 2: Apply the Midpoint Formula

Upon identifying the coordinates, apply the midway formula which is \((\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2})\). Substituting the values, we get \((\frac{10 + 2}{2}, \frac{4 + 6}{2})\).

Step 3: Calculate the Midpoint

Calculate the midpoint by simplifying the expression obtained in Step 2. Therefore, the midpoint is \((\frac{12}{2}, \frac{10}{2}) = (6, 5)\).

Key Concepts

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, is a branch of mathematics that involves the study of geometric figures through a coordinate system. This approach combines algebra and geometry to allow for the precise and analytical description of geometric shapes. In the context of finding the midpoint of a line segment, coordinate geometry provides an effective way of using algebraic expressions to calculate the exact position of the midpoint based on the coordinates of the segment's endpoints.

When dealing with coordinate geometry, understanding the Cartesian plane is vital. This plane is divided into four quadrants by a horizontal axis, known as the x-axis, and a vertical axis, called the y-axis. Points on this plane are represented by ordered pairs \(x, y\), which denote their location relative to both axes. The midpoint formula is a fundamental tool in coordinate geometry, allowing us to find the center point between two coordinates with efficiency and precision.

Algebraic Expressions

An algebraic expression is a mathematical phrase that can include numbers, variables, and arithmetic operations. It represents a value that can vary depending on the variables involved. In using algebra to solve geometric problems, these expressions provide a clear, concise way to manipulate the rules of arithmetic to find the desired outcome.

For example, in determining the midpoint of a line segment, algebraic expressions are used within the midpoint formula to represent the calculations needed. The expression \(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\) is an algebraic representation of the process we follow to find the midpoint. It demonstrates how we average the x-coordinates of the endpoints and the y-coordinates of the endpoints to arrive at the center point. This is a clear indication of the power of algebraic expressions in translating a geometric concept into a form that is easily calculable.

Line Segments

A line segment is a part of a line that is bounded by two distinct endpoints and contains every point on the line between these endpoints. Unlike lines, which extend infinitely in both directions, line segments have a fixed length. In geometry, the properties of line segments are fundamental for constructing shapes, determining distances, and performing various calculations.

The process of finding a midpoint directly relates to the concept of line segments, as it determines the exact center point that divides the segment into two equal lengths. This point is crucial in various geometrical constructions and proofs. The ability to find the midpoint of a line segment by using the midpoint formula is a practical application of coordinate geometry that blends both the algebraic expressions and the geometric understanding of line segments.