Question

It (2, –2) and (5, 2) are opposite vertices of a square, then the length of the side of the square is ___________.

Answer 1

Length of the opposite vertices of a square = Length of the diagonal

$$\begin{gathered} \therefore \mathrm{Length}\mathrm{of}\mathrm{the}\mathrm{diagonal}=\sqrt{{\left(5-2\right)}^2+{\left(2-\left(-2\right)\right)}^2} \\ =\sqrt{{\left(3\right)}^2+{\left(2+2\right)}^2} \\ =\sqrt{{\left(3\right)}^2+{\left(4\right)}^2} \\ =\sqrt{9+16} \\ =\sqrt{25} \\ =5 \end{gathered}$$

But, Length of the diagonal = $\sqrt{2}$ (Length of the side of the square)

$$\begin{gathered} \therefore 5=\sqrt{2}\times \mathrm{Length}\mathrm{of}\mathrm{the}\mathrm{side} \\ \Rightarrow \mathrm{Length}\mathrm{of}\mathrm{the}\mathrm{side}=\frac{5}{\sqrt{2}} \\ =\frac{5}{\sqrt{2}}\times \frac{\sqrt{2}}{\sqrt{2}} \\ =\frac{5\sqrt{2}}{2} \end{gathered}$$

Hence, the length of the side of the square is $\underline{\frac{5\sqrt{2}}{2}\mathrm{units}}.$