How To Find The Distance Between Two Points

How do you define the distance present between two points? To find an accurate measurement, you need to connect these two points. Both the points need to be in the XY plane and measure the distance by connecting their coordinates. 

By joining the specific coordinates of these two given points, you get a line that connects both the points. The measurement of this line gives you the exact distance between these two points. 

Let’s check the formula used to measure this distance.

What Is Meant By The Distance Formula?

In mathematics, you define the distance formula for the coordinate geometry. Sometimes, you might get slightly confused with the speed and distance used in physics. 

However, calculating the distance between two points can be useful for varied topics as it gives a concrete value of the space between both points. 

To calculate the distance formula, let’s take two points, A (x1,y1) and B (x2,y2), lying in the XY plane. Then the distance formula will be:

D= √[(x2 – x1)2 + (y2 – y1)2]

where D is the distance between A and B points.

How To Determine The Distance Formula Using A Graph?

In the above example, you need to draw two straight lines parallel to the X and Y-axis through A and B. The parallel lines from A (x1,y1) and B (x2,y2) meet at C(x1, y2). Therefore, a triangle is formed, ABC, in which,

BC= base of the triangle

AC=perpendicular on the base BC

AB= Hypotenuse of the right-angled triangle ABC

According to the Pythagorean Theorem,

AB2= AC2 + BC2

Or, AB= √[(x2 – x1)2 + (y2 – y1)2]

Therefore, the distance between these two points, A and B, is √[(x2 – x1)2 + (y2 – y1)2]

Let’s check some examples to use this formula to find the distance between A and B:

Example 1

What is the distance present between A (2,4) and B (-4,4)?

Solution

According to the question, the coordinates of the given points are:

A= (2,4)

B= (-4,4)

Here, x1=2

y1=4

x2= -4

y2= 4

With the help of the distance formula, let’s calculate the distance between these points:

D= Distance between A and B

Or, D= √[(x2 – x1)2 + (y2 – y1)2]

Or, AB= D= √[(x2 – x1)2 + (y2 – y1)2]

Or, AB= √[(-4 – 2)2 + (4 – 4)2]

Or, AB= √[(-6)2 + (0)2]

Or, AB= 6 units.

Therefore, the distance present between A and B is 6 units.  

Example 2

Can you prove that the points (– 4, 4), (4, 2), (1, 7), and (–1, –1) are the vertices of a square? 

Solution

Let’s consider A (1, 7), B(4, 2), C (–1, –1), and D (– 4, 4) as the coordinates of four given points in an XY plane. We need to prove that ABCD is a square. 

To prove that ABCD is a square, we need to provide

AB=BC=CD=DA (as the sides of a square are equal to each other)

AC=BD (as the diagonals of the square are equal in length)

With the help of distance formula,

We can find AB, BC, CD, DA respectively.

AB = √[(1 – 4)2 + (7-2)2] = √(9 + 25) = √34

BC = √[(4+1)2 + (2+1)2] = √(25 + 9) = √34

CD = √[(–1+ 4)2+ (–1 – 4)2] = √(9 + 25) = √34

AD = √[(1+4)2 + (7 – 4)2] = √(25 + 9) = √34

AC =√[(1+1)2+(7+1)2] = √(4+64) = √68

BD = √[(4+4)2 + (2-4)2] = √(64+4) = √68

Since, AB = BC = CD = DA and AC = BD.

Therefore, we can conclude that ABCD or the given points together form a square.

Example 3

Can you find a point on the Y-axis that will be equidistant from both A (2, 3) and B (-1, 2)? 

Solution

Let’s consider C (0,y) to be the required point on the Y-axis that will be equidistant from both A (2, 3) and B (-1, 2)

Therefore, we have to prove that 

PA = PB = PA² = PB²
Or, (2 – 0)² + (3 – y)² = (-1 – 0)² + (2 – y)²

Or, 4 + 9 + y² – 6y = 1 + 4 + y² – 4y

Or, – 6y + 4y = 1 – 9

Therefore, y=4. Hence the number on the y-axis that will be equidistant from both A (2, 3) and B (-1, 2) is (0, 4).