Distance Formula

Distance formula

The Cartesian coordinates are used to represent points in a plane. Every point in a place has a one to one relationship to a coordinate point. The distance (d)  between two points A(x1, y1) and B(x2, y2) is given by the formula
d = √[(x2 − x1)²+(y2 − y1)²]

The above formula can be found with the help of Pythagoras theorem. Draw a right angled triangle as shown in the graph. When we draw a right angled triangle we draw it such that the two edges including the right angle are parallel to the two axis. This helps us to find the coordinate of the vertex. Name the right angled vertex as C. The coordinate of the point C is (x2, y1). a line parallel to the coordinate axis has its other coordinate same. Suppose the line is parallel to x-axis then all the points on the line has its y-coordinate equal.  The distance between the points A and B is d. The distance between the points A and C is AC =(x2 − x1) and distance between the points B and C is BC =(y2 − y1). By Pythagoras Theorem we have (AB)² = (AC)² + (BC)². Substituting the value of AC and BC we get

d² = (x2 − x1)²+(y2 − y1
Taking square root and as distance cannot be negative, we get
d = √[(x2 − x1)²+(y2 − y1)²]

Example:
Distance between A(3,5) and B(1,2) is
AB = d
d = √[(1 − 3)²+(2 − 5)²]
= √(4+9)
= √13

A point in three dimensional geometry has the following formula to find distance
The distance between points A(x1,y1,z1) and B(x2,y2,z2) = √[(x2 - x1)2 + (y2 - y1)2 + (z2 - z1)2]




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