Coordinate Geometry

Distance Formula

You want to go to Bangkok from Mumbai for a concert of your favourite artist, but you don’t know the distance you have to travel. These cities are just 2 points on a map with specific latitudes and longitudes. So how do you calculate the distance between the two points? That’s easy. You use the distance formula. Here’s how we do it.

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What is distance formula?

Derived from the Pythagorean Theorem, the distance formula is used to find the distance between any 2 given points. These points are usually crafted on an x-y coordinate plane.

The formula is,

AB=√[(x2-x1)²+(y2-y1)²]

Let us take a look at how the formula was derived.

Derivation of formula

In the above diagram, let’s consider 2 points A(x1,y1) and B(x2,y2). Connect the points A and B directly forming a slanting line AB. Now complete the triangle by joining the points A and B to a common point C such that AC is parallel to the y-axis while BC is parallel to the x-axis.

Now for the coordinates of point C.

Since we considered the point C to be parallel to x-axis the y-coordinate of C will be the same as the y-coordinate of B which is y2 and since C is also parallel to the y-axis the x-coordinate will be equal to the x-coordinate of A which is x1.

Therefore the coordinates of point C are C (x1,y2).

It is a well-known fact that on the x-y coordinate plane the x-axis cuts the y-axis at 90 degrees. That means both the axes are perpendicular to each other. Since the sides, AC and BC are parallel to the y-axis and the x-axis respectively, what we now have is a right-angled triangle ACB where side AB is the hypotenuse, side AC is the perpendicular and CB is the base.

In order to find the length of side AC, we need to find the distance between points A and C. As we have already established the fact that AC is parallel to the y-axis, the x-coordinates will be the same and we cannot use it to calculate the distance.

The only information we are left with are the y-coordinates. If we observe carefully, subtracting the y-coordinate values of C from the y-coordinate value of A we get the distance between the points A and C.

And that my friends, is how we found the length of side AC.

side AC= y2-y1        (1)

Similarly, we also find the length of side CB. The only difference here will be that the y-coordinates remain the same and we subtract the x-coordinates. Therefore,

side CB= x2-x1       (2)

Triangle ACB is a right-angled triangle. We need to find hypotenuse AB.

To do that we use the Pythagoras Theorem. So,

AB²=BC²+AC²

From (1) and (2),

AB²=(x2-x1)²+(y2-y1)²

Taking Square roots on both sides,

AB=√{(x2-x1)²+(y2-y1)²}

What we derived just now is the Distance formula.

So the Distance Formula is,

AB=√[(x2-x1)²+(y2-y1)²]

Remember, as there is a plus sign in between both the squared values, we cannot take them out of the square root without first performing the addition operation.

Solved Examples

Q1: The coordinates of point A are (-4,0) and the coordinates of point B are (0,3). Find the distance between these two points.

Sol: Coordinates of A = (-4,0) = (x1,y1)

Coordinates of B = (0,3) = (x2,y2)

By Distance Formula,

AB = √{(x2-x1)²+(y2-y1)²} = √{[0-(-4)]²+ (3-0)²}

= √(4²+3²)} = √(16+9) = √25

= 5 units

Hence, according to the Distance Formula, the distance between points A and B is 5 units

Q2: The coordinates of point A are (1,7) and the coordinates of point B are (3,2). Find the distance between these two points.   

Sol: Coordinates of A = (1,7) = (x1,y1)

Coordinates of B = (3,2) = (x2,y2)

By Distance Formula,

AB = √{(x2-x1)²+(y2-y1)²}

= √{(1-3)²+ (7-2)²}

= √{(-2)²+(5)²}

= √(4+25) = √29 units

According to the Distance Formula, the distance between points A and B is √29 units.

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Derivation of section formula in all four quadrants

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