Chapter 12.6, Problem 9P

To determine

To determine the perimeter and the area.

Answer to Problem 9P

Perimeter is 45.44 km

Area is $103.8{\text{ km}}^2$

Explanation of Solution

Given information :

Diagonal paths are 12km and 20km.

We shall use the property of parallelogram that the diagonals bisect each other.

  

Here we shall use the cosine rule.

Cosine formula states that,

  $\cos A=\frac{b^2+c^2-a^2}{2bc}$

Put the values, we get,

  $\begin{array}{l}\cos {60}^{\circ }=\frac{{\left(10\right)}^2+{\left(6\right)}^2-B{C}^2}{2\left(10\right)\left(6\right)}\\ 60=136-B{C}^2\\ B{C}^2=76\\ BC=8.72\end{array}$

Similarly we find other side.

  $\begin{array}{l}\cos {120}^{\circ }=\frac{{\left(10\right)}^2+{\left(6\right)}^2-D{C}^2}{2\left(10\right)\left(6\right)}\\ -60=136-D{C}^2\\ D{C}^2=196\\ DC=14\end{array}$

Hence the perimeter is 2(14+8.72)=45.44km

Now we calculate the area of each triangle and then add them.

We shall use the formula $ab\sin \theta$ for the area of triangle where $a$ and $b$ are two sides and $\theta$ is angle between them.

Hence we get,

  $2\left(\left(10\right)\left(6\right)\sin {120}^{\circ }\right)+2\left(\left(6\right)\left(10\right)\sin {60}^{\circ }\right)=103.8$