How do you find the distance between two points?

Answer 1

For two coordinates points (x1, y1) and x2, y2), you can find the straight line distance using the Pythagorean theorem.

The vertical difference (y1-y2) forms one side of the triangle, and the horizontal difference is the other (x2-x1). The hypotenuse is the straight distance along the line, and is defined by :

h = square root of (a2 plus b2) = square root of [ (y2-y1)2 + (x2-x1)2 ]

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EXAMPLE :

For points (1, 3) and (4, 7), the distance along y is (7-3) and along x is (4-1) and

square root [ 42 + 32 ] = sq rt [9 + 16] = sq rt [25] = 5.

Measuring the distance along the line would verify that the distance is 5.

Answer 2

The question is not well specified and so there are several possible answers.

In n-dimensional Euclidean space, if the coordinates of the two points are A = (a1, a2, ... an) and B = (b1, b2, ... bn), then the minimum distance AB is given by

AB2 = (a1-b1)2 + (a2-b2)2 + ... + (an-bn)2

In 2-dimensional space, this reduces to AB2 = (a1-b1)2 + (a2-b2)2

On a spherical surface, such as the surface of the earth, the shortest distance between two points is along the arc of the great circle: a circle whose centre is the centre of the sphere and whose circumference passes through the two points.

The shortest distance on the surface of a polyhedron may be found by finding the straight line distance between the images of the two points on its net.

Finally, there are other ways of measuring distance. One that is fairly easy to understand, is the Minkovski or taxicab metric (or the Manhattan metric). If you want the distance between two points in a city it is not much use to be told the distance "as the crow flies" - unless you happen to be a crow. What you want is "go x blocks North and then y blocks East": that is a distance of x + y blocks rather than sqrt(x2 + y2) blocks!

Answer 3

(-20,14) and (12,14)