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 coordinate - 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 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 a correct form of the distance formula.
• Option B: $d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}$ is a correct form of the distance formula.
• Option C: $d=\sqrt{(x_2 + x_1)^2+(y_2 + y_1)^2}$ is not the distance formula.
• Option D: $d=\sqrt{|x_2 - x_1|^2+|y_2 - y_1|^2}$ is a correct form as $|a|^2=a^2$.
• Option E: $d=\sqrt{(x_2 + x_1)-(y_2 + y_1)}$ is not the distance formula.

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