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The Distance Formula is a crucial mathematical tool for determining the distance between two points on a coordinate plane. It comes from the Pythagorean theorem, which helps find the hypotenuse of a right triangle. In a two-dimensional space, the distance between points \(x_1, y_1\) and \(x_2, y_2\) is calculated using:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
This formula is derived by considering two points as forming the hypotenuse of a triangle on the Cartesian plane.

When applying the Distance Formula, it's important to:

• Correctly substitute the coordinates in place of \(x_1, y_1\) and \(x_2, y_2\).
• Follow the order of subtraction as indicated in the formula.
• Carry out the arithmetic carefully, especially when working with negative numbers, to avoid mistakes.

Using this formula provides you with a precise measure of the distance between points, which is essential in solving many geometric problems like classifying triangles.

Coordinates are the backbone of geometry on a plane, allowing us to precisely identify points and analyze shapes like triangles. A coordinate system is essentially a grid with an x-axis and a y-axis. Each point is defined by:

• An x-coordinate, which tells you how far to move horizontally from the origin (0,0).
• A y-coordinate, which tells you how far to move vertically from the origin.

These coordinates are usually written as (x, y).

In problems like triangle classification, having a clear understanding of how to read and use coordinates is vital. Coordinates enable us to:

• Compute distances between points using the Distance Formula.
• Draw accurate graphical representations of geometric shapes.
• Apply other geometric concepts such as the slope of a line or the midpoint formula.

Understanding coordinates and being able to manipulate them is key in excelling in geometry-related topics.

Triangles are three-sided polygons, and they can be classified in two primary ways: by their angles or by the lengths of their sides. Here, we focus on classification by side lengths. In this system, there are three main types:

• Equilateral Triangle: All three sides are of equal length. This uniformity in side length means that all internal angles are also equal, each measuring 60 degrees.
• Isosceles Triangle: Two sides are of equal length, while the third is different. The angles opposite the equal sides are also equal.
• Scalene Triangle: All three sides are of different lengths, which means all the angles are also different.

In the given problem, after calculating the lengths of sides using the Distance Formula, we found that none of the sides of \(\triangle ABC\) are equal. This means \(\triangle ABC\) is classified as a scalene triangle. Understanding these classifications helps in not only identifying triangles but also in solving angle and area-related problems in geometry.