Question

Prove that the diagonals of a rectangle are perpendicular if and only if the rectangle is a square.

Answer 1

Let ABCD be a rectangle. Take A as the origin.

Suppose the position vectors of points B and D be $\overrightarrow{a}$ and $\overrightarrow{b}$, respectively.

Now,

$\overrightarrow{\mathrm{AC}}=\overrightarrow{\mathrm{AB}}+\overrightarrow{\mathrm{BC}}=\overrightarrow{\mathrm{AB}}+\overrightarrow{\mathrm{AD}}=\overrightarrow{a}+\overrightarrow{b}$

Also, $\overrightarrow{\mathrm{BD}}=\overrightarrow{a}-\overrightarrow{b}$

Since ABCD is rectangle, so $\overrightarrow{\mathrm{AB}}\perp \overrightarrow{\mathrm{AD}}$.

$\therefore \overrightarrow{a}.\overrightarrow{b}=0$ .....(1)

Now, diagonals AC and BD are perpendicular

iff $\overrightarrow{\mathrm{AC}}.\overrightarrow{\mathrm{BD}}=0$

$$\begin{gathered} \mathrm{iff}\left(\overrightarrow{a}+\overrightarrow{b}\right).\left(\overrightarrow{a}-\overrightarrow{b}\right)=0 \\ \mathrm{iff}{\left|\overrightarrow{a}\right|}^2-{\left|\overrightarrow{b}\right|}^2=0 \\ \mathrm{iff}\left|\overrightarrow{a}\right|=\left|\overrightarrow{b}\right| \\ \mathrm{iff}\left|\overrightarrow{\mathrm{AB}}\right|=\left|\overrightarrow{\mathrm{AD}}\right| \end{gathered}$$

iff ABCD is a square

Thus, the diagonals of a rectangle are perpendicular if and only if the rectangle is a square.