Chapter 4.8, Problem 47E

Solution

To find: The length AC and the length BD .

The length AC and the length BD are approximately 33.6 in. and 12.9 in. respectively.

Given information:

Given, a rhombus is a quadrilateral with all sides equal in length. A rhombus is also a parallelogram.

  

Calculation:

In a rhombus the diagonals are perpendicular bisector to each other and bisect the angles. Here, let the diagonals meet at point O .

The figure representing the data is:

  

Using the trigonometric function definitions,

  $\begin{array}{c}\cos A=\frac{\text{adjacent side}}{\text{hypotenuse}}\\ \cos \left(21°\right)=\frac{AO}{18}\\ AO=18\left(0.934\right)\\ AO\approx 16.8\end{array}$

Hence, $\begin{array}{c}AC=2AO\\ =2\left(16.8\right)\\ =33.6\end{array}$

Also,

  $\begin{array}{c}\sin A=\frac{\text{opposite side}}{\text{hypotenuse}}\\ \sin \left(21°\right)=\frac{BO}{18}\\ BO=18\left(0.358\right)\\ BO\approx 6.45\end{array}$

Hence,

  $\begin{array}{c}BD=2BO\\ =2\left(6.45\right)\\ =12.9\end{array}$

The length AC and the length BD are approximately 33.6 in. and 12.9 in. respectively.

Conclusion:

The length AC and the length BD are approximately 33.6 in. and 12.9 in. respectively.