Chapter 3.7, Problem 49PPS

To determine

To calculate: Area of the triangle formed by the coordinates $(0,3),\text{}\left(4,7\right),(5,9)$

Answer to Problem 49PPS

Area of the triangle $=2$ meters

Explanation of Solution

Given information: coordinates of triangle are $(0,3),\text{}\left(4,7\right),(5,9)$

Formula used: If coordinates of the triangle are $(x_1,y_1),(x_2,y_2),(x_3,y_3)$ then area of triangle is equal to $\frac{1}{2}\left|\begin{array}{ccc}{x}_1& y_1& 1\\ x_2& y_2& 1\\ x_3& y_3& 1\end{array}\right|$

Calculation:

Let $(x_1,y_1)=(0,3),(x_2,y_2)=(4,7),(x_3,y_3)=(5,9)$

Now area of the triangle is given by $\frac{1}{2}\left|\begin{array}{ccc}{x}_1& y_1& 1\\ x_2& y_2& 1\\ x_3& y_3& 1\end{array}\right|$

Put values of $(x_1,y_1),(x_2,y_2),(x_3,y_3)$ .

So,

Area of the triangle $=\frac{1}{2}\left|\begin{array}{ccc}0& 3& 1\\ 4& 7& 1\\ 5& 9& 1\end{array}\right|$

Now find the value of $\frac{1}{2}\left|\begin{array}{ccc}0& 3& 1\\ 4& 7& 1\\ 5& 9& 1\end{array}\right|$

  $\frac{1}{2}\left|\begin{array}{ccc}0& 3& 1\\ 4& 7& 1\\ 5& 9& 1\end{array}\right|=\frac{1}{2}\left[0(7-9)-3(4-5)+1(36-35)\right]=\frac{1}{2}\left(0+3+1\right)=\frac{4}{2}=2$

So,

Area of the triangle $=2$ meters