How does the distance formula show the Pythagorean Theorem?
The distance formula is used to measure the distance between two points in a coordinate plane. It is derived from the Pythagorean Theorem, which 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.
In the distance formula, if we have two points A(x1, y1) and B(x2, y2), the distance between them can be calculated using the formula:
√((x2 - x1)^2 + (y2 - y1)^2)
This formula can be seen as a direct application of the Pythagorean Theorem, where the horizontal and vertical distances between the two points represent the lengths of the two sides of a right triangle, and the diagonal distance represents the length of the hypotenuse. The formula essentially finds the length of the hypotenuse using the Pythagorean Theorem.