Question

The diagonal of the given square is 4 cm, its perimeter and area will be equal to

• A. $8\sqrt{2}cm,4c{m}^2$
• B. $4\sqrt{2}cm,8c{m}^2$
• C. $8\sqrt{2}cm,8c{m}^2$
• D. $4\sqrt{2}cm,4c{m}^2$

Answer 1

The correct option is C

$8\sqrt{2}cm,8c{m}^2$

Angles of the triangle ABC are ${45}^{\circ },{45}^{\circ },{90}^{\circ }$

$\Rightarrow \sin \left(45\right):\sin \left(45\right):\sin \left(90\right)$

$\Rightarrow \frac{1}{\sqrt{2}}:\frac{1}{\sqrt{2}}:1$

$\Rightarrow 1:1:\sqrt{2}$

So, the sides AB,BC & AC will be in the ratio $1:1:\sqrt{2}$

Hence $AB:BC:AC\equiv 1:1:\sqrt{2}$

The ratio will be $x:x:\sqrt{2}x$

Since AC =4, So $\sqrt{2}x=4$

$\Rightarrow x=2\sqrt{2}$

So, the ratio AB:BC:AC will be equal to $2\sqrt{2}:2\sqrt{2}:4$

which gives $AB=2\sqrt{2}cm$

$BC=2\sqrt{2}cm$

AC =4 cm

Area $=AB\times BC=2\sqrt{2}\times 2\sqrt{2}=8c{m}^2$

Perimeter $=4\left(AB\right)=4\times 2\sqrt{2}=8\sqrt{2}cm$