Question

select all the equations that represent the distance formula.
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. $d = sqrt{|x_2 - x_1|^2+|y_2 - y_1|^2}$
e. $d = sqrt{(x_2 + x_1)-(y_2 + y_1)}$

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 given by $d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}$. 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 valid. 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 also 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 distance - formula structure.
• Option D: $d=\sqrt{|x_2 - x_1|^2+|y_2 - y_1|^2}$ is correct because of the property $|a|^2=a^2$.
• Option E: $d=\sqrt{(x_2 + x_1)-(y_2 + y_1)}$ is incorrect as it does not follow the distance - formula structure.

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. $d=\sqrt{|x_2 - x_1|^2+|y_2 - y_1|^2}$