Problem 96 - Algebra A Combined Function

Question

Find the midpoint of each line segment whose endpoints are given. \((\sqrt{8},-\sqrt{12}) ;(3 \sqrt{2}, 7 \sqrt{3})\)

Step-by-Step Solution

Answer

The midpoint is \( \left( \frac{5\sqrt{2}}{2}, \frac{5\sqrt{3}}{2} \right) \).

Step 1: Midpoint Formula Introduction

The formula for finding the midpoint \( M \) of a line segment with endpoints \((x_1, y_1)\) and \((x_2, y_2)\) is \( M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \). We will use this formula to find the midpoint of the given endpoints.

Step 2: Assign Values to Variables

Assign the given endpoint values to the variables: \(x_1 = \sqrt{8}\), \(y_1 = -\sqrt{12}\), \(x_2 = 3\sqrt{2}\), and \(y_2 = 7\sqrt{3}\).

Step 3: Calculate the Midpoint X-Coordinate

Using the formula from Step 1, substitute the values into the formula for the x-coordinate: \[ \frac{\sqrt{8} + 3\sqrt{2}}{2} = \frac{2\sqrt{2} + 3\sqrt{2}}{2} = \frac{5\sqrt{2}}{2} \].

Step 4: Calculate the Midpoint Y-Coordinate

Now, using the formula, substitute the values into the formula for the y-coordinate: \[ \frac{-\sqrt{12} + 7\sqrt{3}}{2} = \frac{-2\sqrt{3} + 7\sqrt{3}}{2} = \frac{5\sqrt{3}}{2} \].

Step 5: Write the Midpoint

Combine the results from Steps 3 and 4 to write the midpoint \( M \): \( M = \left( \frac{5\sqrt{2}}{2}, \frac{5\sqrt{3}}{2} \right) \).

Key Concepts

Line Segment

A line segment is a straight path that connects two points in space. Unlike a line, which continues indefinitely in both directions, a line segment has a definite starting and ending point. These two points are known as endpoints.
Line segments are a fundamental building block in geometry, helping us to describe shapes and dimensions. Calculating midpoints of line segments is a common practice, especially when dealing with geometry problems. To visualize a line segment, think of a tight rope stretched between two poles.
The entire rope from one pole to the other represents the line segment, with the poles being the endpoints. In coordinate geometry, we often use a plane to set precise locations for these endpoints, allowing us to use formulas to find various properties like length or midpoint.

Endpoints

Endpoints define the beginning and end of a line segment, making them crucial for understanding and measuring the segment itself. In the given exercise, the endpoints are:

• First endpoint: \( (\sqrt{8}, -\sqrt{12}) \)
• Second endpoint: \( (3\sqrt{2}, 7\sqrt{3}) \)

These coordinates indicate exact positions in a coordinate plane, featuring numbers sometimes expressed as radicals.Endpoints allow us to use various formulas to explore geometric properties. For instance, they'd help in calculating the length of the line segment or finding the midpoint using the midpoint formula.
Understanding endpoints is crucial in geometry as they set limits and provide reference points for calculations. They are also important when visualizing the geometric shapes and their locations on the Cartesian plane.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, provides a bridge between algebra and geometry through graphs and equations. It revolutionizes how geometric problems are solved, allowing for precise calculations and visual representations.
In coordinate geometry, shapes are plotted using a coordinate plane consisting of an x-axis and a y-axis. These axes divide the plane into four quadrants, each providing a unique way to describe positions in space.The ability to pinpoint exact locations with coordinates simplifies complex problems tremendously.
For example, the endpoint \( (\sqrt{8}, -\sqrt{12}) \) is positioned based on its x and y coordinates. So, rather than relying solely on visual estimation, coordinate geometry enables the use of formulas, like the midpoint formula, to solve problems with precision.This method has a wide application range from basic geometrical problems to advanced real-life situations involving navigation, engineering, and physical sciences.

Radicals

Radicals, commonly known as roots, are mathematical symbols used to represent the root of a number. The most familiar radical is the square root, indicated by the symbol \(\sqrt{}\). In the given exercise, radicals feature prominently in defining endpoints:

• \(\sqrt{8}\) and \(3\sqrt{2}\) for x-coordinates
• \(-\sqrt{12}\) and \(7\sqrt{3}\) for y-coordinates

Radicals can sometimes make geometric calculations more complex, as they need to be simplified before using them in formulas like the midpoint formula.
For instance, \(\sqrt{8}\) simplifies to \(2\sqrt{2}\), and \(\sqrt{12}\) becomes \(2\sqrt{3}\).Handling radicals is crucial to achieve accurate solutions in coordinate geometry and other fields requiring precise numerical computations.
Despite their complexity, radicals help maintain more exact representations of numbers, as rounding might lead to errors in precise calculations.Learning to manipulate radicals effectively is a useful skill in many areas - from mathematics to engineering and beyond.