Problem 43 - College Algebra with Modeling and Visualization

Question

Find the midpoint of the line segment connecting the points. $$ (\sqrt{2}, \sqrt{5}),(\sqrt{2},-\sqrt{5}) $$

Step-by-Step Solution

Answer

The midpoint is \((\sqrt{2}, 0)\).

Step 1: Identify Midpoint Formula

The midpoint of a line segment connecting two points \((x_1, y_1)\) and \((x_2, y_2)\) in the coordinate plane is calculated using the formula: \[\left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \].

Step 2: Substitute the Values

We are given the points \((\sqrt{2}, \sqrt{5})\) and \(\sqrt{2}, -\sqrt{5})\). Substitute \(x_1 = \sqrt{2}\), \(y_1 = \sqrt{5}\), \(x_2 = \sqrt{2}\), and \(y_2 = -\sqrt{5}\) into the midpoint formula. This gives us:\[\left( \frac{\sqrt{2} + \sqrt{2}}{2}, \frac{\sqrt{5} + (-\sqrt{5})}{2} \right) \].

Step 3: Simplify the Midpoint Calculation

Calculate the x-coordinate: \[\frac{\sqrt{2} + \sqrt{2}}{2} = \frac{2\sqrt{2}}{2} = \sqrt{2} \]Calculate the y-coordinate: \[\frac{\sqrt{5} + (-\sqrt{5})}{2} = \frac{0}{2} = 0 \].

Step 4: Write the Final Midpoint

The midpoint of the line segment is \[(\sqrt{2}, 0) \].

Key Concepts

Coordinate Plane

The coordinate plane is a fundamental concept in mathematics that helps us understand the positioning of points in a two-dimensional space. It is also known as the Cartesian plane, named after the mathematician René Descartes. The coordinate plane is composed of two perpendicular lines: the x-axis (horizontal) and the y-axis (vertical). These axes intersect at a point known as the origin, which is denoted as \((0, 0)\).On this plane, any point is represented by an ordered pair \((x, y)\), where \(x\) is the horizontal position and \(y\) represents the vertical position. By understanding the coordinate plane, you can plot points to visualize and solve problems related to geometry. The plane is split into four quadrants, labeled from I to IV, allowing us to determine the positive or negative values of \(x\) and \(y\). This division helps in identifying the location of a point precisely. Use this system to explore relationships, calculate distances, or even determine midpoints as in the given exercise.

Coordinate Geometry

Coordinate geometry, often called analytic geometry, combines algebra and geometry to solve problems. It allows us to analyze geometric figures using coordinates and equations. By plotting points on the coordinate plane, you can investigate distances between points, slopes of lines, and other critical properties.For example, finding the midpoint of a line segment is a common task. The midpoint formula, \[\left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)\]helps us locate the exact center between two points. This formula gives us insight into symmetry and geometric configurations.Coordinate geometry not only helps in solving straight-line problems but also aids in understanding curves. It is a powerful tool for both theoretical studies and practical applications, ranging from simple mappings to complex navigational systems.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operations that represent mathematical ideas. They're a foundational component in algebra, used to convey quantities and relationships.In coordinate geometry, algebraic expressions help us define and manipulate points and lines. For instance, the expression \((\sqrt{2}, \sqrt{5})\)uses the square root function to describe coordinates. By treating coordinates as algebraic expressions, we can use substitution, simplification, and evaluation to solve problems, such as finding midpoints or distances.These expressions allow for the symbolic manipulation necessary to solve real-world problems, abstracting complex relationships into understandable forms. Mastering algebraic expressions is crucial for interpreting and solving a wide array of mathematical challenges, especially within the context of coordinate geometry.