Question

what is the distance formula?

$\boldsymbol{\text{option a: } sqrt{(x_2 + x_1)^2 - (y_2 + y_1)^2}}$

$\boldsymbol{\text{option b: } sqrt{(x_2 - x_1) + (y_2 - y_1)}}$

$\boldsymbol{\text{option c: } sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}}$

$\boldsymbol{\text{option d: } sqrt{(y_2 - y_1)^2 - (x_2 - x_1)^2}}$

Explanation:

The distance formula between two points $(x_1,y_1)$ and $(x_2,y_2)$ in a coordinate plane is derived from the Pythagorean theorem. The horizontal distance between the points is $|x_2 - x_1|$ and the vertical distance is $|y_2 - y_1|$. By the Pythagorean theorem, the distance $d$ is $\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}$, which matches option C. Options A, B, and D have incorrect operations (addition/subtraction instead of the correct squared terms and sum) in the formula.

Answer:

C. $\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$