Question

which of the following represent the distance formula? select all that apply. a. $d = sqrt{(x_1 - x_2)^2+(y_1 - y_2)^2}$ b. $d = sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}$ c. $d = sqrt{(x_2 + x_1)^2+(y_2 + y_1)^2}$ d. $d = sqrt{|x_2 - x_1|^2+|y_2 - y_1|^2}$

Explanation:

Step1: Recall distance formula

The distance $d$ between two points $(x_1,y_1)$ and $(x_2,y_2)$ in a two - dimensional plane is based on the Pythagorean theorem. It is $d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}$. Also, since $(x_1 - x_2)^2=(x_2 - x_1)^2$ and $(y_1 - y_2)^2=(y_2 - y_1)^2$, $d=\sqrt{(x_1 - x_2)^2+(y_1 - y_2)^2}$ is also correct. And $|x_2 - x_1|^2=(x_2 - x_1)^2$ and $|y_2 - y_1|^2=(y_2 - y_1)^2$, so $d = \sqrt{|x_2 - x_1|^2+|y_2 - y_1|^2}$ is correct.

Step2: Analyze each option

• Option A: $d=\sqrt{(x_1 - x_2)^2+(y_1 - y_2)^2}$ is correct as explained above.
• Option B: $d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}$ is the standard form of the distance formula.
• Option C: $d=\sqrt{(x_2 + x_1)^2+(y_2 + y_1)^2}$ is incorrect as it does not follow the Pythagorean - based distance formula.
• Option D: $d=\sqrt{|x_2 - x_1|^2+|y_2 - y_1|^2}$ is correct because $|a|^2=a^2$ for any real number $a$.

Answer:

A. $d = \sqrt{(x_1 - x_2)^2+(y_1 - y_2)^2}$, B. $d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}$, D. $d=\sqrt{|x_2 - x_1|^2+|y_2 - y_1|^2}$