Question

$△ABC$ is a triangle with vertices $A=(x,x+3),B=(2,1)$ and $C=(3,-2)$. Area of the triangle is $5$.

Coordinates of the centroid of $△ABC$ are

• A. $(\frac{7}{2},\frac{13}{2})$
• B. $(\frac{17}{6},\frac{11}{6})$
• C. $(\frac{5}{2},\frac{3}{2})$
• D. $(\frac{19}{6},\frac{21}{6})$

Answer 1

The correct option is B $(\frac{17}{6},\frac{11}{6})$

Area of the triangle as per given co-ordinates is:-

$\frac{1}{2}\left|\begin{array}{ccc}x& x+3& 1\\ 2& 1& 1\\ 3& -2& 1\end{array}\right|=5$

$\Rightarrow |x\left(3\right)-(x+3)(-1)+1(-7)|=10$

$\Rightarrow |3x+x+3-7|=10$

$\Rightarrow |4x-4|=10\Rightarrow 4x=10+4$

$\Rightarrow 4x=14$

$\Rightarrow x=\frac{7}{2}$

Hence, centroid of triangle is,
$(\frac{\frac{7}{2}+2+3}{3},\frac{\frac{13}{2}+1-2}{3})=(\frac{17}{6},\frac{11}{6})$