Question

What is the formula to find distance in coordinate geometry, given coordinates (x_1, y_1), (x_2, y_2)? (valid variables for this problem: x_1, x_2, y_1, y_2)

Answer 1

Answer:

The distance formula is: $d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$

Step-by-step explanation:

Given:

Two points on a coordinate plane are given as:

$(x_1,y_1)\ and\ (x_2,y_2)$

Now, let the points be $A(x_1,y_1)\ and\ B(x_2,y_2)$ 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 = $d$

AC = $x_2-x_1$

BC = $y_2-y_1$

Now, we use Pythagoras theorem to find 'd'. This gives,

$AB^2=AC^2+BC^2\\\\d^2=(x_2-x_1)^2+(y_2-y_1)^2$

Now, taking square root on both sides, we get:

$\sqrt{d^2}=\pm\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}\\\\d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$

Ignoring the negative result as distance can never be negative.

Therefore, the formula  to find distance in coordinate geometry, given coordinates $(x_1, y_1), (x_2, y_2)$ is:

$d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$