Chapter 8, Problem 131RE

To determine

Find the area of the given triangle using vertices.

Answer to Problem 131RE

The area of the triangle is $16$ square units.

Explanation of Solution

Given information:

The given triangle is

  

Calculation:

The area of a triangle with vertices

  $\left(x_1,y_1\right)$ ,

  $\left(x_2,y_2\right)$ and $\left(x_3,y_3\right)$ is

Area $=\pm \frac{1}{2}\left|\begin{array}{ccc}{x}_1& y_1& 1\\ x_2& y_2& 1\\ x_3& y_3& 1\end{array}\right|$

Where the symbol $\pm$ indicates that the appropriate sign should be chosen to yield a positive area.

In the given triangle $\begin{array}{l}\left(x_1,y_1\right)=\left(1,0\right)\\ \left(x_2,y_2\right)=\left(5,8\right)\\ \left(x_3,y_3\right)=\left(5,0\right)\end{array}$

Now,

Area $=\pm \frac{1}{2}\left|\begin{array}{ccc}1& 0& 1\\ 5& 8& 1\\ 5& 0& 1\end{array}\right|$

Find the value of determinant.

  $\begin{array}{l}\left|\begin{array}{ccc}1& 0& 1\\ 5& 8& 1\\ 5& 0& 1\end{array}\right|\\ =1\left(8\cdot 1-0\cdot 1\right)-0\left(5\cdot 1-5\cdot 1\right)+0\left(5\cdot 0-5\cdot 8\right)\\ =8-0-40\\ =-32\end{array}$

Area $=-\frac{1}{2}\left(-32\right)=16$ [chose

  $(-)$ so that the area is positive)

Hence thearea of the triangle is $16$ square units.