Question

The side $AB$ of a square $ABCD$ is parallel to the $x-\text{axis}$. Find the slopes of all its sides.
Also find:
i) The slope of diagonal $AC$
ii) The slope of diagonal $BD$

Answer 1

Finding the slope of all sides of squareWe know that, the slope of any line parallel to $x-\text{axis}$ is $0$.
$\therefore \text{Slope of}AB=0$
As, $CD\parallel AB$,
$\Rightarrow \text{Slope of}CD=\text{Slope of}AB=0$

Now,
$BC\perp AB$ (Adjacent sides of square)
We know that,
if two lines are perpendicular then product of their slope is $-1$.

$\Rightarrow \text{Slope of}BC=\frac{-1}{\text{slope of}AB}$

$=\frac{-1}{0}=\text{not defined}$.

Similarly,
$AD\perp AB$ (Adjacent sides of square)

We know that, if two lines are perpendicular then product of their slope is $-1$

$\Rightarrow \text{Slope of}AD=\frac{-1}{\text{slope of}AB}$

$=\frac{-1}{0}=\text{not defined}$.

Finding the slope of the diagonals of a square
(i) Slope of diagonal $AC$

The diagonal $AC$ makes an angle of ${45}^{\circ }$ with the positive direction of $x-\text{axis}$,

$\Rightarrow \text{Slope of}AC=tan{45}^{\circ }=1$

(ii) Slope of diagonal $BD$

The diagonal $BD$ makes an angle of $-{45}^{\circ }$ with the positive direction of $x-\text{axis}$,

$\Rightarrow \text{Slope of}BD=tan{45}^{\circ }=1$

Hence, slope of side $AB\text{and}CD$ is $0$,
slope of side $BC\text{and}AD$ is not defined.
Slope of diagonals $AC\text{and}BD$ is $1$ and $-1$ respectively.