Question

The centroid of a triangle lies at the origin and the coordinates of its two vertices are $\left(-8,7\right)$ and $\left(9,4\right)$. The area of the triangle is

• A. $\frac{95}{6}$
• B. $\frac{-285}{2}$
• C. $\frac{190}{3}$
• D. $285$

Answer 1

The correct option is B

$\frac{-285}{2}$

Explanation for the correct option:

Finding the centroid:

Centroid of the triangle formed by vertices $(x_1,y_1),(x_2,y_2),(x_3,y_3)$ is given by

$x=\frac{x_1+x_2+x_3}{3}$ and $y=\frac{y_1+y_2+y_3}{3}$

Here, the centeroid is $\left(0,0\right)$ and two vertices are $\left(-8,7\right)$ and $\left(9,4\right)$.

Substitute the values.

$$\begin{gathered} \therefore 0=\frac{-8+9+x_3}{3} \\ \Rightarrow x_3=-1 \end{gathered}$$

and

$$\begin{gathered} 0=\frac{7+4+y_3}{3} \\ \Rightarrow y_3=-11 \end{gathered}$$

So, the vertices for triangle are $\left(-8,7\right),\left(-1,-11\right)$ and $\left(9,4\right)$.

Area of triangle with three vertices is $\frac{1}{2}\left|\begin{array}{ccc}{x}_1& y_1& 1\\ x_2& y_2& 1\\ x_3& y_3& 1\end{array}\right|$.

$$\begin{gathered} \therefore Area=\frac{1}{2}\left|\begin{array}{ccc}-8& 7& 1\\ 9& 4& 1\\ -1& -11& 1\end{array}\right| \\ =\frac{1}{2}\left[-8\left(4+11\right)-7\left(9+1\right)+1\left(-99+4\right)\right] \\ =\frac{1}{2}\left[-120-70-95\right] \\ =\frac{-285}{2} \end{gathered}$$

Therefore, the correct answer is option (B).