Problem 29 - College Algebra Essentials

Question

find the midpoint of each line segment with the given endpoints. $$ (\sqrt{18},-4) \text { and }(\sqrt{2}, 4) $$

Step-by-Step Solution

Answer

The midpoint of the line segment is \((\sqrt{10}, 0)\)

Step 1: Identify the coordinates

First, label the given coordinates for clarity. Let Point 1 be \((\sqrt{18}, -4)\) and Point 2 be \((\sqrt{2}, 4)\). So, \(x1 = \sqrt{18}, y1 = -4, x2 = \sqrt{2}, y2 = 4\).

Step 2: Apply the midpoint formula

Next, apply the midpoint formula. So, the x-coordinate of the midpoint will be \((\sqrt{18}+\sqrt{2})/2\) and the y-coordinate will be \((-4+4)/2\).

Step 3: Calculate the midpoint

Perform the operations to find the actual coordinates. The x-coordinate of the midpoint will be \((\sqrt{18}+\sqrt{2})/2 = \sqrt{10}\) and the y-coordinate will be \((-4+4)/2 = 0\).

Key Concepts

The Midpoint Formula

Understanding the midpoint formula is essential for finding the exact middle point of a line segment in coordinate geometry. This formula is expressed as \(M_x = \frac{x_1 + x_2}{2}\) and \(M_y = \frac{y_1 + y_2}{2}\), where \(M_x\) and \(M_y\) are the x and y coordinates of the midpoint, and \(x_1, y_1\) and \(x_2, y_2\) are the coordinates of the endpoints of the line segment.

To apply this formula accurately, one must carefully substitute the given endpoint coordinates into the equation. Once the coordinates are added, simply divide by two to locate the midpoint. Through this process, we ensure the middle point is equally distant from both endpoints along the line segment. Utilizing the midpoint formula provides a straightforward method for solving many problems in coordinate geometry and is a foundational tool for students studying algebra and geometry.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, creates a link between algebra and geometry through the use of a coordinate system. This allows for geometric problems to be solved algebraically, and vice versa. When given a pair of coordinates, such as \( (x_1, y_1)\) and \( (x_2, y_2)\), we can graph these points on the coordinate plane and find important information about the line segment joining these points.

In the case of finding midpoints, the coordinate plane provides a visual representation to understand how the midpoint formula actually reflects the halfway mark between two points. The concepts of the x-axis (horizontal line) and the y-axis (vertical line) are instrumental when identifying the location of points and thereby applying the midpoint formula effectively. Coordinate geometry is a versatile and powerful mathematical tool used extensively in several fields, including physics, engineering, and computer graphics.

Radicals in Algebra

Radicals, which are an expression for the root of a number, frequently occur in algebraic equations. They may seem daunting at first, but understanding how to simplify and manipulate radicals is crucial, especially when dealing with coordinate geometry. For example, \(\sqrt{18}\) can be simplified to \(3\sqrt{2}\) because 18 is 9 times 2, and the square root of 9 is 3. Therefore, the radical simplifies the numbers under the root to make equations more manageable.

In the case of finding the midpoint of a line segment, the presence of radicals in the coordinates may initially complicate the arithmetic. However, by simplifying the radicals as much as possible before applying the midpoint formula, the computation becomes much clearer. Handling radicals correctly is a key skill in algebra and is crucial for solving problems in a more efficient and simplified manner, paving the way for a stronger grasp of mathematics as a whole.