Question

if the length of one side of a rhombus is 6 cm and the angle is 60 degree then the area of a rhombus is ​

Answer 1

SOLUTION:-

°.° Diagonal bisect the opposite angles

$\therefore \angle OAB = \frac{ \cancel{ 60 \degree} }{ \cancel{ 2}} = 30 \degree$

$in \: right \: \angle AOB$

$\sin30 \degree = \frac{OB}{AB } = > \frac{1}{2} = \frac{OB}{6}$

$= > OB = \frac{ \cancel{6}}{ \cancel{2} } = 3cm$

$\therefore BD = 2 O B = 2 \times 3 = 6cm$

$and \: \cos30 \degree = \frac{AO} { OB } = \frac{ \sqrt{3} }{2} = \frac{AO}{6}$

$= > AO = \frac{ \cancel{ 6} \sqrt{3} }{ \cancel{ 2} } = 3 \sqrt{3} cm$

$\therefore AC = 3 \sqrt{3} \times 2 = 6 \sqrt{3} cm$

$\therefore diagonals \: are \: 6 \: and \: 6 \sqrt{3} cm$

we know that area of rhombus is,

$= \frac{1}{2} ( d_{1} \times d_{2}) \\ = \frac{1}{ \cancel{2}} ( \cancel{ 6} \times 6 \sqrt{3} ) \\ = (3 \times 6 \sqrt{3} ) \\ = 18 \sqrt{3} {cm}^{2}$

$hence \: area \: of \: rhombus \: is \: \boxed{18 \sqrt{3} {cm}^{2} }$

Answer 2

refer to the attachment