Problem 60 - Elementary and Intermediate Algebra

Question

Find the midpoint of the line segment with the given endpoints. \((-5,-2),(7,3)\)

Step-by-Step Solution

Answer

The midpoint is \((1, \frac{1}{2})\).

Step 1: Identify the Formula

To find the midpoint of a line segment, we use the formula for the midpoint \((x_m, y_m)\), which is \((\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2})\), where \((x_1, y_1)\) and \((x_2, y_2)\) are the coordinates of the endpoints.

Step 2: Substitute the Values

Plug the given endpoints \((-5,-2)\) and \((7,3)\) into the formula: \(x_1 = -5\), \(y_1 = -2\), \(x_2 = 7\), \(y_2 = 3\). Substitute these into the midpoint formula: \((\frac{-5 + 7}{2}, \frac{-2 + 3}{2})\).

Step 3: Calculate the Midpoint

Calculate each coordinate of the midpoint: \(x_m = \frac{-5 + 7}{2} = \frac{2}{2} = 1\) and \(y_m = \frac{-2 + 3}{2} = \frac{1}{2}\).

Step 4: State the Midpoint

Combine the coordinates to form the midpoint, which is \((1, \frac{1}{2})\).

Key Concepts

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, is a branch of mathematics that uses geometric figures to showcase algebraic relationships through a coordinate system. Its primary function is to link the algebra of equations with geometric figures by employing ordered pairs, often denoted as \((x, y)\). This powerful tool allows us to understand geometric shapes in the two-dimensional plane and solve geometric problems using algebraic formulas.
Imagine a graph with an x-axis and a y-axis intersecting at a point called the origin, labeled \((0,0)\). Every point in the plane is represented by a unique \((x, y)\) coordinate, indicating the location of a point concerning the origin. This coordinate system is helpful in many aspects of mathematics, such as solving for distances and midpoints.

• Points: Represented as \((x, y)\) coordinates.
• Lines: Linear equations in the form \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept.
• Distance: Determined using the distance formula between two points.

Coordinate geometry extends beyond these basics, providing tools to analyze circles, ellipses, parabolas, among others by enabling their representation through algebraic equations.

Line Segment

A line segment is a part of a line that is bounded by two distinct endpoints, and it contains every point on the line between its endpoints. In coordinate geometry, you can visualize a line segment as the shortest path between two points \((x_1, y_1)\) and \((x_2, y_2)\). This segment is key in various calculations, such as finding the distance between two points or identifying the midpoint of the segment.
Midpoint calculation is one of the most common tasks involving line segments. This involves finding a point that divides the segment into two equal lengths or halves. The midpoint provides us with a way to evenly split the segment or find a balance point, which can be crucial in geometric designs, construction, and further geometric exploration.
More formally, the midpoint \((x_m, y_m)\) is the average of the x-coordinates and the y-coordinates of the endpoints:

• X-coordinate of Midpoint: \(x_m = \frac{x_1 + x_2}{2}\)
• Y-coordinate of Midpoint: \(y_m = \frac{y_1 + y_2}{2}\)

Understanding line segments helps enrich knowledge on how various geometric calculations are interrelated.

Algebraic Calculation

Algebraic calculation is a process of using mathematical operations to solve equations or evaluate expressions. In coordinate geometry, algebraic calculations help us determine valuable information such as distances, midpoints, and even slopes. These calculations rely on algebraic formulas that translate geometric meanings into numeric results.
For instance, when we find the midpoint of a line segment, we apply the midpoint formula, which uses simple arithmetic operations (addition and division) to compute the answer:

• Add the x-coordinates: \((-5 + 7)\)
• Divide by 2: \(\frac{2}{2} = 1\)
• Add the y-coordinates: \((-2 + 3)\)
• Divide by 2: \(\frac{1}{2}\)

The resulting coordinates, \((1, \frac{1}{2})\), are the numeric representation of the midpoint. Algebra serves as the backbone of coordinate geometry, connecting known formulas with a strong understanding of arithmetic operations to solve problems effectively and efficiently. This promotes precise and meaningful interpretations which are pivotal in mathematical explorations and everyday applications.