91Ó°ÊÓ

In coordinate geometry, finding the midpoint of a line segment is a valuable concept. This involves averaging the coordinates of two points, effectively finding the middle. To determine the midpoint between two points \(A (x_1, y_1)\) and \(B (x_2, y_2)\), use the formula:

• \( M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \)

This formula calculates the average of the x-coordinates and the y-coordinates separately.
For example, in our solution, if the coordinates are \( (6\sqrt{5}, 2\sqrt{3}) \) and \( (2\sqrt{5}, 3\sqrt{3}) \,\) the midpoint \( M \) is calculated as follows:
First, average the x-coordinates: \( \frac{6\sqrt{5} + 2\sqrt{5}}{2} = 4\sqrt{5} \).
Then, average the y-coordinates: \( \frac{2\sqrt{3} + 3\sqrt{3}}{2} = \frac{5\sqrt{3}}{2} \).
Therefore, the midpoint \( M \) of the line segment connecting these two points is \( (4\sqrt{5}, \frac{5\sqrt{3}}{2}) \). This demonstrates the beauty of symmetry when working with coordinates.

Simplifying radicals is a crucial skill in mathematics, particularly when dealing with expressions involving square roots. The goal of simplifying is to express a number in its simplest radical form, which often aids in calculation and comparison.

• For instance, to simplify \( 2\sqrt{45} \), recognize that 45 can be broken down into perfect squares \( \text{(i.e., } 45 = 9 \times 5) \).
• Thus, \( 2\sqrt{45} = 2\times3\sqrt{5} = 6\sqrt{5} \).

This process involves:

• Identifying factors that are perfect squares.
• Taking the square root of these factors and simplifying accordingly.

Let's look at another example:

• \( \sqrt{12} = \sqrt{4 \times 3} = 2\sqrt{3} \).
• Similarly, \( \sqrt{20} = 2\sqrt{5} \) and \( \sqrt{27} = 3\sqrt{3} \).

By simplifying radicals, you can perform operations and calculations more efficiently, keeping numbers more manageable.

Coordinate geometry, or analytic geometry, is a fascinating field that connects algebra and geometry. It allows for the representation of geometric shapes and figures using a coordinate system. One significant aspect of coordinate geometry is working with the distance formula, which helps find the distance between two points in a plane.

• The distance formula is derived from the Pythagorean theorem: \( d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2} \).

This formula measures the Euclidean distance between two points \( (x_1, y_1) \) and \( (x_2, y_2) \).
In the example given, by substituting the points after simplifying the radicals, the distance was calculated as:

• \( d = \sqrt{(-4\sqrt{5})^2 + (\sqrt{3})^2} = \sqrt{80 + 3} = \sqrt{83} \).

Coordinate geometry provides the tools required to compute these distances and, moreover, to investigate the properties of shapes such as lines, circles, and parabolas using a simple and efficient algebraic approach.
It lays the foundational groundwork that connects abstract algebraic concepts to concrete geometric interpretations.