Chapter 8.5, Problem 73E

To determine

Find the area of triangle.

Answer to Problem 73E

  $\boxed{12}$ Sq units.

Explanation of Solution

Given information:

Use determinants to find the area of a triangle with vertices $\left(3,-1\right),\left(7,-1\right),$ and $\left(7,5\right)$ . Confirm your answer by plotting the points in a coordinate plane and using the formula

Area $=\frac{1}{2}\left(base\right)\left(height\right)$

Calculation:

We have to find out the area of a triangle with vertices $\left(3,-1\right),\left(7,-1\right),\left(7,5\right)$ .

The area of a triangle whose vertices are given as $\left(x_1,y_1\right),\left(x_2,y_2\right),\left(x_3,y_3\right)$ can be found out using the formula:

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

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

Using the above formula we get,

  $\frac{1}{2}\left|\begin{array}{ccc}{x}_1& y_1& 1\\ x_2& y_2& 1\\ x_3& y_3& 1\end{array}\right|=\frac{1}{2}\left|\begin{array}{ccc}3& -1& 1\\ 7& -1& 1\\ 7& 5& 1\end{array}\right|$

  $=\frac{1}{2}\left[24\right]$

  $=12$

Hence the area of triangle is given by

  $+\frac{1}{2}\left|\begin{array}{ccc}{x}_1& y_1& 1\\ x_2& y_2& 1\\ x_3& y_3& 1\end{array}\right|=+12$

  $=\boxed{12}$ sq units.

Let us consider that $A(3,-1),B(7,-1),C(7,5)$ , we can draw these points as

  

From the graph we can see that the height of the triangle is $6$ units and the length of the base is $4$ units. Now the area of triangle is

Area $=\frac{1}{2}\left(base\right)\left(height\right)$

  $=\frac{1}{2}\left(4\right)\left(6\right)$

  $=\boxed{12}$ sq units.

Hence, the area of triangle $=\boxed{12}$ sq units.