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Coordinate Geometry – Distance Between Two Points or Distance Formula

Distance Between Two Points or Distance Formula-amansmathsblogs.com

Distance Between Two Points or Distance Formula


We need to find the distance between two points on Rectangular Coordinate Plane. For this, take two points in XY plane as P and Q whose coordinates are P(x1, y1) and Q(x2, y2). We join P and Q and make a right triangle PQR as shown in the figure below.

Distance Between Two Points or Distance Formula

Thus, we need to find PQ. From the figure, we have PR = (x2-x1) and QR  = (y2-y1). 

In right triangle PQR, we use Pythagoras Theorem as

PQ2 = PR2 + QR2 = (x2-x1)2 + (y2-y1)2 \Rightarrow PQ = \sqrt{ \left ( x_{2}-x_{1} \right )^{2}+ \left ( y_{2}-y_{1} \right )^{2}}

Thus, the distance between two points P(x1, y1) and Q(x2, y2) is

PQ =  \sqrt{ \left ( x_{2}-x_{1} \right )^{2}+ \left ( y_{2}-y_{1} \right )^{2}}

Some Important Points:

1. If O is the origin and P(x, y) is any point, then from distance formula OP = \sqrt{ x ^{2}+ y^{2}}

2. For three points to be collinear, the sum of the distance between two pairs of points is equal to the third pair of the points. 

3. Given the two points (x1y1) and (x2y2), they only indicate that there is a “first” point and a “second” point; that is, that you have two points. Whichever one you call “first” or “second” is up to you. The distance will be the same, regardless.

4. The most common mistake made when using the Formula is to accidentally mismatch the x-values and y-values. Be careful you don’t subtract an x from a y, or vice versa; make sure you’ve paired the numbers properly.

See this video about How To Find the Distance Between Two Points:


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