Answer:
The distance formula is: ![d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}](https://tex.z-dn.net/?f=d%3D%5Csqrt%7B%28x_2-x_1%29%5E2%2B%28y_2-y_1%29%5E2%7D)
Step-by-step explanation:
Given:
Two points on a coordinate plane are given as:
![(x_1,y_1)\ and\ (x_2,y_2)](https://tex.z-dn.net/?f=%28x_1%2Cy_1%29%5C%20and%5C%20%28x_2%2Cy_2%29)
Now, let the points be
and let 'd' be the distance between the two points.
Now, join the points A and B to make a line segment AB.
Also, draw a right angled triangle ABC right angled at point C. Point C is is the intersection of horizontal and vertical lines drawn from points A and B respectively as shown in the figure below.
From the triangle ABC, we observe that:
AB = ![d](https://tex.z-dn.net/?f=d)
AC = ![x_2-x_1](https://tex.z-dn.net/?f=x_2-x_1)
BC =
Now, we use Pythagoras theorem to find 'd'. This gives,
![AB^2=AC^2+BC^2\\\\d^2=(x_2-x_1)^2+(y_2-y_1)^2](https://tex.z-dn.net/?f=AB%5E2%3DAC%5E2%2BBC%5E2%5C%5C%5C%5Cd%5E2%3D%28x_2-x_1%29%5E2%2B%28y_2-y_1%29%5E2)
Now, taking square root on both sides, we get:
![\sqrt{d^2}=\pm\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}\\\\d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}](https://tex.z-dn.net/?f=%5Csqrt%7Bd%5E2%7D%3D%5Cpm%5Csqrt%7B%28x_2-x_1%29%5E2%2B%28y_2-y_1%29%5E2%7D%5C%5C%5C%5Cd%3D%5Csqrt%7B%28x_2-x_1%29%5E2%2B%28y_2-y_1%29%5E2%7D)
Ignoring the negative result as distance can never be negative.
Therefore, the formula to find distance in coordinate geometry, given coordinates
is:
![d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}](https://tex.z-dn.net/?f=d%3D%5Csqrt%7B%28x_2-x_1%29%5E2%2B%28y_2-y_1%29%5E2%7D)