Question

The perimeter of a rhombus is 96 cm and the obtuse angle of it is ${120}^{\circ }$.

Find the length of its smallest diagonal.

Answer 1

Let ABCD be the rhombus with perimeter = $96$

Let $\angle ABC={120}^{\circ }$

Then, $\angle CDA=\angle ABC=120$ (Opposite angles)

and, $\angle C=\angle A=x$

Sum of angles of rhombus = 360

$\angle A+\angle B+\angle C+\angle D=360$

$2x=360-240$

$x={60}^{\circ }$

Now, all sides of rhombus are equal. hence, length of the side = $\frac{96}{4}$

= $24$ cm

In $△ABC$

$\cos B=\frac{A{B}^2+B{C}^2-A{C}^2}{2\times AB\times BC}$

$\cos 120=\frac{{24}^2+{24}^2-A{C}^2}{2\times 24\times 24}$

$\frac{-1}{2}=\frac{{24}^2+{24}^2-A{C}^2}{2\times 24\times 24}$

$-{24}^2=2\times {24}^2-A{C}^2$

$AC=24\sqrt{3}$

$AC=41.56$ cm

In $△ABD$

$\angle A={60}^{\circ }$

$\cos A=\frac{A{B}^2+A{D}^2-B{D}^2}{2\times AB\times AD}$

$\cos 60=\frac{{24}^2+{24}^2-B{D}^2}{2\times 24\times 24}$

$\frac{1}{2}=\frac{2\times {24}^2-B{D}^2}{2\times 24\times 24}$

${24}^2=2\times {24}^2-B{D}^2$

$BD=24$ cm