2.2
The Distance Formula
Two friends enter a park from different entrances and stop at different corners of a field.
From their respective positions, they want to determine the exact distance between them.
The distance formula is applied to calculate the distance between the two positions, marked as points A and B, using their coordinates.
It finds the shortest path between them—like cutting diagonally across a field instead of walking around the edges.
This calculation is based on the Pythagorean Theorem, where the horizontal and vertical distances meet at Point C, forming a right triangle.
The theorem states that the square of the hypotenuse, or the diagonal side, equals the sum of the squares of the other two sides.
On a coordinate grid, these sides represent the differences in the x- and y-values between points A and B.
By squaring the horizontal and vertical differences, adding them together, and taking the square root, the length of the hypotenuse is determined.
This relationship gives the straight-line distance between the two friends.
Such calculations are also used in mapping routes and planning wireless network coverage.
In geometry, measuring the direct distance between two points on a plane is essential in various practical and theoretical applications. Whether in navigation, engineering, or computer graphics, determining the shortest path between two locations involves using the distance formula. This formula is derived from the Pythagorean Theorem, which relates the lengths of the sides of a right triangle. On a coordinate plane, the horizontal and vertical distances between two points serve as the legs of a right triangle, and the line connecting the points forms the hypotenuse. By calculating the square root of the sum of the squares of these horizontal and vertical differences, the straight-line distance can be accurately determined.
A city planner is mapping two key locations in a town: a library at L (300, 500) and a sports complex at S (1000, 1200). To determine how far these locations are apart, the distance formula is used:
Substituting the given coordinates:
Thus, the library and the sports complex are approximately 990 meters apart.
This method illustrates how geometric principles are practically applied in urban planning, allowing for accurate distance measurement essential for designing efficient layouts and infrastructure.
Two friends enter a park from different entrances and stop at different corners of a field.
From their respective positions, they want to determine the exact distance between them.
The distance formula is applied to calculate the distance between the two positions, marked as points A and B, using their coordinates.
It finds the shortest path between them—like cutting diagonally across a field instead of walking around the edges.
This calculation is based on the Pythagorean Theorem, where the horizontal and vertical distances meet at Point C, forming a right triangle.
The theorem states that the square of the hypotenuse, or the diagonal side, equals the sum of the squares of the other two sides.
On a coordinate grid, these sides represent the differences in the x- and y-values between points A and B.
By squaring the horizontal and vertical differences, adding them together, and taking the square root, the length of the hypotenuse is determined.
This relationship gives the straight-line distance between the two friends.
Such calculations are also used in mapping routes and planning wireless network coverage.
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