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The distance formula is a useful tool in geometry for determining the distance between two points on a plane. Given two points with coordinates

• \((x_1, y_1)\)
• \((x_2, y_2)\)

we use the formula:\[\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\]This formula is based on the Pythagorean Theorem. It essentially finds the hypotenuse of a right triangle formed by the horizontal and vertical distances between the points.
For example, to find the distance between points \(A(-3,2)\) and \(B(1,0)\), calculate:

• Horizontal distance: \(1 - (-3) = 4\)
• Vertical distance: \(0 - 2 = -2\)

The distance then becomes: \[\sqrt{4^2 + (-2)^2} = \sqrt{16 + 4} = \sqrt{20} = 2\sqrt{5}\]Using the distance formula simplifies problems involving distances between any two points in a coordinate plane, which are crucial in many geometric proofs.

The Pythagorean theorem is a fundamental principle in geometry. It asserts that in a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. This relationship is expressed by the equation:\[c^2 = a^2 + b^2\]Consider triangle \(\Delta ABC\), where the sides are measured as follows:

• \(AB = 2\sqrt{5}\)
• \(BC = 3\sqrt{5}\)
• \(CA = \sqrt{65}\)

To check if it is a right triangle, identify the longest side, which is assumed to be the hypotenuse. Here, \(CA\) is the longest side:

• Calculate \((2\sqrt{5})^2 + (3\sqrt{5})^2 = 20 + 45 = 65\)
• Compare it to \((\sqrt{65})^2 = 65\)

Since these values match, the theorem confirms that \(\triangle ABC\) is right-angled. Understanding how to apply the Pythagorean theorem in various contexts helps solve many geometric problems, especially those involving distance and angles.

Coordinate geometry, also known as analytic geometry, connects algebra and geometry using a coordinate system. This fusion allows us to solve geometric problems algebraically. A coordinate system represents points using pairs of numbers containing coordinates such as

• \((x,y)\)

which specify a point's position horizontally and vertically.

Using this method, we can calculate distances, midpoints, and more using algebraic formulas. In our exercise:

• Points \(A, B, C\) have coordinates \(A(-3,2)\), \(B(1,0)\), and \(C(4,6)\).
• This setup helps us to apply the distance formula to find lengths of triangle sides \(AB\), \(BC\), and \(CA\).
• Both the distance formula and the Pythagorean Theorem are rooted in coordinate geometry.

This approach simplifies finding properties like slopes and shapes' angles and can be applied to vectors, circles, and more complex curves. Embracing coordinate geometry enhances problem-solving skills in various scientific and engineering disciplines.