Chapter 8.4, Problem 29WE

To determine

To find: The lengths of the diagonals of the rhombus.

Answer to Problem 29WE

  $16\sqrt{3}\text{ and 16}$

Explanation of Solution

Given information: The perimeter of a rhombus is 64 and one of its angles has measure 120.

Calculation:

The perimeter of the rhombus is 64.

Then, the length of each side $=\frac{64}{4}=16$

Let the lengths of the diagonals be x and y.

Since, the diagonals of rhombus bisect each other at right angles.

Then, using Trigonometric ratios,

  $\begin{array}{l}\sin 60°=\frac{\text{opposite}}{\text{hypotenuse}}\\ \Rightarrow \sin 60°=\frac{\frac{x}{2}}{16}\\ \Rightarrow \sin 60°=\frac{x}{32}\\ \Rightarrow x=32\times \sin 60°[\because \sin 60°=\frac{\sqrt{3}}{2}]\\ \Rightarrow x=32\times \frac{\sqrt{3}}{2}\\ \Rightarrow x=16\sqrt{3}\end{array}$

Again,

  $\begin{array}{l}\cos 60°=\frac{\text{adjacent}}{\text{hypotenuse}}\\ \Rightarrow \cos 60°=\frac{\frac{y}{2}}{16}\\ \Rightarrow \cos 60°=\frac{y}{32}\\ \Rightarrow y=32\times \cos 60°\\ \Rightarrow y=32\times \frac{1}{2}[\because \cos 60°=\frac{1}{2}]\\ \Rightarrow y=16\end{array}$

Thus, the lengths of the diagonals are $16\sqrt{3}\text{ and 16}$ .