Question

If the extremities of a diagonal of a square are $(1,-2,3)$ and $(2,-3,5)$, then the length of its side $\left(\text{in units}\right)$ is:

• A. $\sqrt{6}$
• B. $\sqrt{3}$
• C. $\sqrt{5}$
• D. $\sqrt{7}$

Answer 1

The correct option is B $\sqrt{3}$

Let $s$ be the side and $d$ be the diagonal of a square respectively.
Hence $s=\frac{d}{\sqrt{2}}\cdots \left(i\right)$
Let $A(1,-2,3)$ and $C(2,-3,5)$ be the extremities of the diagonal AC
The length of diagonal $d=\sqrt{(x_2-x_1{)}^2+(y_2-y_1{)}^2+(z_2-z_1{)}^2}$
$d=\sqrt{(2-1{)}^2+(-3+2{)}^2+(5-3{)}^2}$
$d=\sqrt{1+1+4}$
$d=\sqrt{6}$
From equation $\left(i\right)$
$s=\frac{d}{\sqrt{2}}=\frac{\sqrt{6}}{\sqrt{2}}=\sqrt{3}$
Length of side $=\sqrt{3}$ units