Chapter 9.7, Problem 18PSB

To determine

To calculate: The Length of each diagonal.

Answer to Problem 18PSB

Length of each diagonal is $6$ and $6\sqrt{3}$ units.

Explanation of Solution

Given information:Here, we have given that one angle of rhombus is ${120}^o$ and perimeter is $24$ .

Formula used: Perimeter of rhombus = $4$ (side)

Calculation: Let ABCD be the rhombus with diagonals AC and BD .

Perimeter of rhombus = $4$ (side)

  $\Rightarrow 24=4$ (side)

  $\Rightarrow$ Side = $\frac{24}{4}=6$ units

As, Diagonals of rhombus bisect the angles of rhombus,

  $\begin{array}{l}\Rightarrow \angle CBO=\frac{1}{2}\times 120\\ \Rightarrow \angle CBO=60\end{array}$

Now, as the angles of triangle BOC are $30°,60°,90°$ ,

Hypotenuse = $6$ = $2(3)$

  $\Rightarrow OB=3$ and $OC=3\sqrt{3}$ (If hypotenuse is $2a$ , then other two sides are $a$ and $a\sqrt{3}$ ).

Now, as diagonals of rhombus bisect each other,

  $\Rightarrow BD=2(OB)=2(3)=6$

And $AC=2(OC)=2(3\sqrt{3})=6\sqrt{3}$