Question

Find the area of the triangle ABC.

$\begin{array}{l}A.\text{10squareunits}\\ \text{B.12}\mathrm{square}\mathrm{units}\\ \text{C.14}\mathrm{square}\mathrm{units}\\ \text{D.16}\mathrm{square}\mathrm{units}\\ \text{E.Noneofthese}\end{array}$

Short answer

The area of the triangle is 14 square units.

Step-by-step solution

Step 1- Check the figure.

Consider the figure.

The vertices from the graph are as follows:

$\begin{array}{l}A(x_1,y_1)=(2,3)\\ B(x_2,y_2)=(1,-3)\\ C(x_3,y_3)=(-3,1)\end{array}$

Step 2- Step description.

The area of the triangle can be calculated by the following formula:

$A=\frac{1}{2}\left[x_1\left(y_2-y_3\right)+x_2\left(y_3-y_1\right)+x_3\left(y_1-y_2\right)\right]$

Step 3- Step description.

Substitute the values in the formula and simplify as follows:

$\begin{array}{l}A=\frac{1}{2}[x_1(y_2-y_3)+x_2(y_3-y_1)+x_3(y_1-y_2)]\\ A=\frac{1}{2}[2(-3-1)+1(1-3)+(-3)(3+3)]\\ A=\frac{1}{2}[2(-4)-2-3\left(6\right)]\\ A=\frac{1}{2}[-8-2-18]\\ A=\frac{1}{2}(-28)\\ A=-14\end{array}$

As area cannot be negative then consider only the numerical value 14.

Therefore, the area of the triangle is 14 square units.