Chapter 8.4, Problem 28WE

To determine

To find: the dimensions of rectangle.

Answer to Problem 28WE

The dimensions of the rectangle are 4 and $4\sqrt{3}$ .

Explanation of Solution

Given:

The diagonals of a rectangle are 8 units long and intersecting at $60°$ .

Calculation:

The rectangle is shown below:

In triangle OCE,

  $\begin{array}{c}\cos \angle EOC=\frac{\text{Base}}{\text{Hypotenuse}}\\ \cos 30°=\frac{OE}{4}\\ \frac{\sqrt{3}}{2}=\frac{OE}{4}\\ OE\approx 2\sqrt{3}\end{array}$

Hence, the side AB is calculated as follows:

  $\begin{array}{c}AB=2FB\left(FB=OE\right)\\ AB=2\left(2\sqrt{3}\right)\\ =4\sqrt{3}\end{array}$

And,

  $\begin{array}{c}\sin \angle EOC=\frac{\text{perpendicular}}{\text{Hypotenuse}}\\ \sin 30°=\frac{CE}{4}\\ \frac{1}{2}=\frac{CE}{4}\\ CE\approx 2\end{array}$

Another side BC of rectangle is calculated as follows:

  $\begin{array}{c}BC=2CE\\ =2\left(2\right)\\ =4\end{array}$

Thus, the dimensions of the rectangle are 4 and $4\sqrt{3}$ .