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Coordinate Geometry is a branch of geometry where points are defined by coordinates on a plane. Each point has two coordinates, known as the x-coordinate and the y-coordinate, which determine its position in a two-dimensional space. Think of a point on this grid as a specific location defined by these numbers.

For our exercise, the points given are \((-2, 6)\) and \((3, -6)\). Here:

• Point 1 has an x-coordinate of -2 and a y-coordinate of 6.
• Point 2 has an x-coordinate of 3 and a y-coordinate of -6.

These coordinates tell us the exact location of each point on the coordinate plane. Understanding how to read these coordinates is a foundation for more complex calculations, such as finding the distance between two points.

The Distance Formula in coordinate geometry helps us calculate how far apart two points are on a plane. It is a straightforward application of the Pythagorean Theorem, transforming it to work with coordinates. The formula is:\[d = \sqrt{ (x_2 - x_1)^2 + (y_2 - y_1)^2 }\]This formula requires arithmetic manipulation to find the difference between the x-coordinates and y-coordinates of the two points and then use these differences to compute the distance.

For our points:

• Substitute \(x_1 = -2, y_1 = 6, x_2 = 3, y_2 = -6\) into the formula.
• Calculate each part: \((x_2 - x_1) = 3 - (-2) = 5\) and \((y_2 - y_1) = -6 - 6 = -12\).
• Square both results: \(5^2 = 25\) and \((-12)^2 = 144\).
• Add them together: \(25 + 144 = 169\).
• Take the square root: \(\sqrt{169} = 13\).

The result gives us the direct Euclidean distance between these two points.

The Pythagorean Theorem is a fundamental principle in geometry, stating that in a right-angled triangle, the square of the length of the hypotenuse \(c\) (the side opposite the right angle) is equal to the sum of the squares of the other two sides \(a\) and \(b\). It is expressed mathematically as:\[c^2 = a^2 + b^2\]This theorem laid the groundwork for the Distance Formula used in coordinate geometry by treating the distance between two points as the hypotenuse of a right triangle.

In our exercise, when we find the differences \((x_2 - x_1)\) and \((y_2 - y_1)\), these differences form the two shorter sides \(a\) and \(b\) of a right-angled triangle. The diagonal or hypotenuse \(c\), which is the distance we calculate, fully utilizes the theorem.

Breaking this down:

• The side "a" is the horizontal difference between the x-coordinates.
• "b" is the vertical difference between the y-coordinates.
• The calculated distance \(d\) is the length of the hypotenuse.

Thus, the Pythagorean Theorem seamlessly aids in understanding how distances between points are measured in the coordinate plane.