Question

The distance 'd' between any two random points on the Cartesian plane $A(x_1,y_1)andB(x_2,y_2)$is:

• A. $\sqrt{{(x_2-x_1)}^2+{(y_2-y_1)}^2}$
• B. $\sqrt{{(x_2-x_1)}^2-{(y_2-y_1)}^2}$
• C. $\sqrt{{(x_2+x_1)}^2+{(y_2+y_1)}^2}$
• D. $\sqrt{{(x+x_1)}^2-{(y_2+y_1)}^2}$

Answer 1

The correct option is A $\sqrt{{(x_2-x_1)}^2+{(y_2-y_1)}^2}$


Using the Pythagorean theorem, we can derive the distance formula between any two points
$A(x_1,y_1)andB(x_2,y_2)$

From the figure, we can see that the two points form a right triangle by vertiCally dropping point $B$ and horizontally extending point $A$.

Let the distance between the two points be $d$ which is nothing but the length of the hypotenuse of the right triangle.

Let $a=(x_2-x_1)$
and $b=(y_2-y_1)$.
Using the Pythagorean theotem,
$d^2=a^2+b^2$
or $d=\sqrt{a^2+b^2}$

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