Problem 95 - Algebra A Combined Function

Question

Find the midpoint of each line segment whose endpoints are given. \((\sqrt{2}, 3 \sqrt{5}) ;(\sqrt{2},-2 \sqrt{5})\)

Step-by-Step Solution

Answer

(\sqrt{2}, \frac{\sqrt{5}}{2})

Step 1: Understanding the Midpoint Formula

The formula for finding the midpoint \[M(x, y) = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)\] is used to calculate the midpoint of a line segment whose endpoints are given by \((x_1, y_1)\) and \((x_2, y_2)\).

Step 2: Identify the Endpoints

Here, the endpoints are \((\sqrt{2}, 3\sqrt{5})\) and \((\sqrt{2}, -2\sqrt{5})\), with \(x_1 = \sqrt{2}, y_1 = 3\sqrt{5}\) and \(x_2 = \sqrt{2}, y_2 = -2\sqrt{5}\).

Step 3: Applying to the X-Coordinates

Using the midpoint formula for the x-coordinates, we have:\[M_x = \frac{\sqrt{2} + \sqrt{2}}{2} = \frac{2\sqrt{2}}{2} = \sqrt{2}\]

Step 4: Applying to the Y-Coordinates

Using the midpoint formula for the y-coordinates, we have:\[M_y = \frac{3\sqrt{5} + (-2\sqrt{5})}{2} = \frac{3\sqrt{5} - 2\sqrt{5}}{2} = \frac{\sqrt{5}}{2}\]

Step 5: Finding the Midpoint

Combine the results from both coordinates:\[M = (\sqrt{2}, \frac{\sqrt{5}}{2})\]This gives us the midpoint of the line segment.

Key Concepts

Line Segment Midpoint

When we talk about finding a midpoint of a line segment, we are essentially aiming to locate the exact center point that lies between two distinct endpoints. This concept is crucial in various areas of geometry and forms the basis for dividing a line segment into two equal parts.

The midpoint can be thought of as "the average" of the coordinates of the endpoints. Imagine if you were walking along a straight path from one endpoint to another. The midpoint is the exact spot you'd reach when you are halfway there.

• It's calculated using the midpoint formula: \[M(x, y) = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)\]
• The formula shows that you find an average for both x-coordinates and y-coordinates.
• This principle applies to both 2D and 3D geometry, meaning it can be applied in higher dimensions by adding additional coordinates.

Understanding this formula is fundamental for solving geometric problems involving line segments, whether on paper or an analytical application.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, allows us to explore geometry using a coordinate system. This approach enables us to use algebraic techniques to solve geometric problems, making geometry calculations more precise and less abstract.

In this coordinate system, points are placed on a plane typically referred to as the Cartesian plane. Here are some essentials of coordinate geometry:

• A point is defined by coordinates, noted as \(x, y\), which determine its exact location on the plane.
• A line segment is the portion between two points described by these coordinates.
• We utilize formulas to calculate distances, midpoints, and slopes associated with these points.

By leveraging this system, we can not only find the midpoint of a line segment as demonstrated, but also derive significant properties of geometric shapes. For instance, coordinate geometry is instrumental in finding the area of polygons, the equation of circles, and much more.

Endpoints Calculation

Calculating endpoints is a basic yet vital task in geometry that allows us to succinctly define the bounds of a line segment. This calculation is pivotal for solving various geometric problems and constructing geometric shapes.

Endpoints are specifically the two points that mark the beginning and end of a line segment. Knowing these enables us to find:

• The length of the line segment using the distance formula.
• The midpoint, to identify the central point of the segment.
• The slope, to understand the angle and direction at which the segment extends.

In problems like finding the midpoint, identifying the endpoints is the first logical step. For example, with variables like \(\sqrt{2}, 3\sqrt{5}\) and \(\sqrt{2}, -2\sqrt{5}\), you clearly label one as \(x_1, y_1\) and the other as \(x_2, y_2\) to apply relevant formulas. This practice establishes the foundation upon which further geometric computation is built.