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The distance formula allows us to determine the distance between two points in a coordinate plane. Understanding this concept is key in proving properties of triangles, like whether a triangle is isosceles. The formula is derived from the Pythagorean theorem and looks like this: \[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \] When applying this formula, just plug in the coordinates of the two points you are comparing. - Imagine you have points \((x_1, y_1)\) and \((x_2, y_2)\). - Subtract the corresponding x-coordinates \((x_2 - x_1)\) and y-coordinates \((y_2 - y_1)\). - Square both results and add them together. - Finally, take the square root of the sum to get the distance.This formula is especially useful because it works in any part of the coordinate plane, whether your points are in positive or negative quadrants.

Coordinate geometry, often called analytic geometry, is the study of geometric figures through the use of a coordinate system. It opens up powerful tools for mathematical analysis and proofs. Here's why it's essential:- It provides a way to describe the position of points using ordered pairs \((x, y)\). - By using a coordinate system, you can easily calculate the distance between points, the slope of a line, and the area of regions. - It allows geometric shapes like triangles to be expressed in algebraic equations.In this exercise, coordinate geometry helps us place the triangle's vertices on a coordinate plane and then use the distance formula to calculate the lengths of sides. This approach can confirm properties, such as checking for congruent sides in triangles or collinearity of points.

Triangles are fundamental shapes in geometry, and understanding their properties is essential. Here, we're focusing on isosceles triangles, which have two sides of equal length. This property leads to several interesting characteristics: - Isosceles triangles have at least two congruent sides. - The angles opposite these equal sides are also equal. - They can be further specialized into equilateral triangles, where all three sides are equal. In the given exercise, by calculating the distances and discovering two equal sides, we confirm the triangle is isosceles. Recognizing these properties allows us to identify and prove isosceles triangles in different contexts.