Chapter 4.5, Problem 3CYP

To determine

To find: The area of the neighbourhood which is a triangle with the given vertices.

Answer to Problem 3CYP

The area of the neighbourhood which is a triangle with the given vertices $\left(3,15\right)$ , $\left(6,4\right)$ and $\left(11,9\right)$ is $\text{17}{\text{.5km}}^2$ .

Explanation of Solution

Given:

The coordinates for the corners of the triangle are given as:

  $\left(3,15\right)$ , $\left(6,4\right)$ and $\left(11,9\right)$ .

Here each unit represents $0.5$ kilometre.

The map of the triangle with coordinates $\left(3,15\right)$ , $\left(6,4\right)$ and $\left(11,9\right)$ is given as:

  

The area of the triangle with vertices $\left(a_1,a_2\right)$ , $\left(b_1,b_2\right)$ and $\left(c_1,c_2\right)$ is given as:

  $\text{Area}=\left|\frac{a_1\left(b_2-c_2\right)+b_1\left(c_2-a_2\right)+c_1\left(a_2-b_2\right)}{2}\right|$

So the area of the neighbourhood which is a triangle with the given vertices $\left(3,15\right)$ , $\left(6,4\right)$ and $\left(11,9\right)$ is calculated as:

  $\begin{array}{c}\text{Area}=\left|\frac{3\left(4-5\right)+6\left(9-15\right)+11\left(15-4\right)}{2}\right|\\ =35\\ \text{Area}=35\times 0.5\left[\text{As}\text{each}\text{unit}=0.5\right]\\ 17.5{\text{km}}^2\end{array}$

Therefore, the area of the neighbourhood which is a triangle with the given vertices $\left(3,15\right)$ , $\left(6,4\right)$ and $\left(11,9\right)$ is $\text{17}{\text{.5km}}^2$ .