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The distance formula is essential in coordinate geometry, as it helps us calculate the distance between two points on the Cartesian plane. Understanding this formula is key to solving many geometric problems.

The distance formula is derived from the Pythagorean theorem. It is expressed as:

• \[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]

Where:

• \((x_1, y_1)\) and \((x_2, y_2)\) are the coordinates of the two points.
• \(d\) represents the distance between these points.

This formula is used to determine the lengths of the sides of a triangle, as seen in calculating the distances of sides \(AB\), \(BC\), and \(AC\) in the example problem. For each pair of vertices, substitute the coordinates into the formula to find the side length. Remember to square the differences of the coordinates and take the square root of their sum.

The Pythagorean theorem is a foundation in geometry, especially for right triangles. It states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides.

Mathematically, this is expressed as:

• \(a^2 + b^2 = c^2\)

Where:

• \(c\) is the hypotenuse
• \(a\) and \(b\) are the other two sides

To determine if three points form a right triangle, calculate the distances between each pair of points to find the side lengths, then apply the Pythagorean theorem. In the solution, the points \(A\), \(B\), and \(C\) create a right triangle because the squared lengths of sides \(BC\) and \(AC\) add up to the square of the length of side \(AB\). This confirms that \(AB\) is indeed the hypotenuse.

Coordinate geometry, or analytic geometry, combines algebra and geometry, offering a way to visualize algebraic equations. It provides tools and methods to solve geometric problems using a coordinate system, typically the Cartesian plane.

In coordinate geometry:

• Points are defined with coordinates \((x, y)\).
• Lines, curves, and shapes can be represented and analyzed using equations and formulas.
• It allows us to easily calculate distances and angles between geometric figures.

In our task, coordinate geometry enables us to verify the properties of the triangle formed by the vertices \(A(-2, 4)\), \(B(-3, -8)\), and \(C(2, 2)\). The use of distances and the Pythagorean theorem in this coordinate plane helps confirm that these points indeed form a right triangle.

Vertices are the corner points of a triangle. In a coordinate plane, specifying the vertices' coordinates is crucial, as it helps in the analysis and calculation of the triangle's properties.

For a triangle with vertices labeled \(A\), \(B\), and \(C\), each point has a coordinate pair \((x, y)\), indicating its location on the Cartesian plane.

Knowing these points:

• Allows calculation of the side lengths using the distance formula.
• Enables verification of whether the triangle is a right triangle using the Pythagorean theorem.
• Helps determine other geometrical properties such as angles or the perimeter.

In the given exercise, specifying the vertices \(A(-2, 4)\), \(B(-3, -8)\), and \(C(2, 2)\) provides all the information needed to determine the side lengths systematically and to apply mathematical theorems accurately. This forms the basis for confirming the type and properties of the triangle.