Question

Points $A\left(3,1\right),B\left(12,-2\right)\mathrm{and}C\left(0,2\right)$ cannot be the vertices of a triangle. State whether the following statement is true or false. Justify your answer.

• A. True
• B. False

Answer 1

The correct option is A True

Determine whether the given statement is true.

The points are given as $A\left(3,1\right),B\left(12,-2\right)\mathrm{and}C\left(0,2\right)$.

Consider the following coordinates:

$\begin{array}{rcl}A\left(3,1\right)& =& \left(x_1,y_1\right)\\ B\left(12,-2\right)& =& \left(x_2,y_2\right)\\ C\left(0,2\right)& =& \left(x_3,y_3\right)\end{array}$

Apply the area of the triangle: $A=\frac{1}{2}\left[x_1\left(y_2-y_3\right)+x_2\left(y_3-y_1\right)+x_3\left(y_1-y_2\right)\right]$

Substitute the values of $\left(x_1,y_1\right),\left(x_2,y_2\right),\mathrm{and}\left(x_3,y_3\right)$ into the formula.

$\begin{array}{rcl}A& =& \frac{1}{2}\left[3\left(-2-2\right)+12\left(2-1\right)+0\left(1-\left(-2\right)\right)\right]\\ & \Rightarrow & A=\frac{1}{2}\left[3\left(-4\right)+12\left(1\right)+0\right]\\ & \Rightarrow & A=\frac{1}{2}\left[-12+12\right]\\ & \Rightarrow & A=0\end{array}$

Since the area is $0$, the given points must be collinear.

Therefore, the given points cannot be the vertices of a triangle.

Hence, the given statement is true.