Chapter 6, Problem 31RE

To determine

To calculate: The length of sides of the parallelogram when the length of the diagonals are 10 feet and 16 feet.

Answer to Problem 31RE

The length of sides of the parallelogram is 4.3 feet and 12.6 feet.

Explanation of Solution

Given information:

The length of the diagonalsof the parallelogram are 10 feet and 16 feet.

Formula Used:

The law of sines formula:

  $\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}$

The law of cosines formula:

  $a^2=b^2+c^2-2bc\cos A$

  $b^2=a^2+c^2-2ac\cos B$

  $c^2=a^2+b^2-2ab\cos C$

Calculation:

Consider the parallelogram $ABCD$ .

The length of the diagonals $AC=10cmandBD=16cm$ .

  $\angle COD=28°$ .

Take $\Delta COD$ . Find the value of $ODandOC$ .

  $\begin{array}{c}OD=\frac{DB}{2}\\ =\frac{16}{2}\\ =8\end{array}$

  $\begin{array}{c}OC=\frac{AC}{2}\\ =\frac{10}{2}\\ =5\end{array}$

In $\Delta COD$ , use cosine formula to find the value of CD.

  $C{D}^2=O{D}^2+O{C}^2-2\left(OD\right)\left(OC\right)\cos 28°$

Substitute 8 for OD, 5 for OC and 0.8829 for $\cos 28°$ .

  $\begin{array}{c}C{D}^2={\left(8\right)}^2+{\left(5\right)}^2-2\left(8\right)\left(5\right)\left(0.8829\right)\\ =64+25-70.635\\ =89-70.635\\ =18.365\end{array}$

Take square root on both sides.

  $\begin{array}{c}\sqrt{C{D}^2}=\sqrt{18.365}\\ CD=4.285\\ =4.3\end{array}$

In $\Delta BOC$ ,

  $\begin{array}{c}\angle BOC=180°-28°\\ =152°\end{array}$

Use cosine formula to find the value of BC.

  $B{C}^2=O{B}^2+O{C}^2-2\left(OB\right)\left(OC\right)\cos 152°$

Substitute 8 for OB, 5 for OC and 0.8829 for $\cos 152°$ .

  $\begin{array}{c}B{C}^2={\left(8\right)}^2+{\left(5\right)}^2-2\left(8\right)\left(5\right)\left(-0.8829\right)\\ =64+25+70.635\\ =89+70.635\\ =159.635\end{array}$

Take square root on both sides.

  $\begin{array}{c}\sqrt{B{C}^2}=\sqrt{159.635}\\ BC=12.63\\ =12.6\end{array}$

Hence, the length of sides of the parallelogram is 4.3 feet and 12.6 feet.