Distance Between Two Points

Coordinates of a Point

Coordinates of a point, the distance between points, and other similar topics are a part of two-dimension geometry. The coordinates of a point is a set of two number that defines the exact location of the point in the coordinate plane. When the point P is in the two-dimensional plane means that the point is some number of units away from the y-axis and some number of units away from the x-axis. 

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The coordinates of a point on the y-axis are given as ( 0, b ) where b is the distance of the point from the origin and the coordinates of a point on the x-axis are given as ( b, 0 ) where b is the distance from the origin of the point from the origin.

Distance Formula

Let us assume that a boy is walking towards the north direction for 20 meters. Then he takes a turn towards the east direction and walks 30 meters in that direction. Now, how do we calculate the shortest path distance the place where he started and the final destination of the boy?

Below is the pictorial representation of the situation that is stated above. 

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Consider the initial point as P and the final point as R and the distance between P and Q is 20 meters and the distance between O and R is 30 meters. Now, you have to find the distance between the point P and R because it is the shortest distance between the initial point and the final point. This is where you would apply the Pythagorean Theorem. The distance from origin formula is:

\[PR^{2}\] = \[PQ^{2}\] + \[QR^{2}\]

PR = \[\sqrt{PQ^{2}+QR^{2}}\]

PR = \[\sqrt{20^{2}+30^{2}}\]

PR = \[\sqrt{400+600}\]

PR = \[\sqrt{1000}\]

PR = \[10\sqrt{10}\] m

Therefore, the distance between the initial point and final point =\[10\sqrt{10}\] m. In a similar manner, using the Pythagoras Theorem or the right angle theorem you can calculate the distance between the two points that are located in the coordinate plane. Let us understand that concept now.

Distance Between a Point From its Origin

In the figure shown below, notice how the point P( x, y ) is in the coordinate plane. Let us understand how to calculate the distance between point P and the origin O. P is y units away from x - axis and x units away from the y - axis.

Using the Pythagoras theorem,

OP2 = x2 + y2

OP = \[\sqrt{x+y}\]

Solved Examples

Question 1: Find the value of k, if the distance between the points P( 4 , -5 ) and Q ( -3 , k ) is 20 units.

Solution: 

By using the distance formula,

The distance between the points ( 4, -5 ) and ( -3, k ) = \[\sqrt{[( - 3 - 4 )^{2}+(k + 5)^{2}}]\] = 20 units

Squaring both sides of the equation gives,

( -7 )2 + ( k + 5 )2 = 400

( k + 5 )2  = 400 – 25 = 375

Taking root on both the sides, we get;

k + 5 = ± \[\sqrt{375}\]

Question 2: Find the value of a, if the distance between the points P ( 1 , - 2 ) and Q ( -3 , k ) is 5 units.

Solution: By distance formula,

Distance between points ( 1, -2 ) and ( -3, k ) = \[\sqrt{[( - 3 -  1 )^{2} + ( k + 2 )^{2}}]\] = 5 units

Squaring both sides of the equation gives,

( -4 )2 + ( k + 2 )2 = 25

( k + 2 )2 = 25 - 16 = 5 

Taking root on both the sides, we get;

k + 2 = ± \[\sqrt{5}\]