Question

If the centroid of a triangle is (1, 4) and two of its vertices are (4, -3) and (-9, 7), then the area of the triangle is

• A. 183 sq. units
• B. $\frac{183}{2}sq.units.$
• C. 366 sq. units
• D. $\frac{183}{4}sq.units.$

Answer 1

The correct option is B

$\frac{183}{2}sq.units.$

Let the given vertices be B and C. Medians AD and BE meet at G. Co-ordinates of G are (1, 4).
Let coordinates of A be (x, y)
So, $\frac{x+4-9}{3}=1and\frac{y-3+7}{3}=4\Rightarrow x-5=3\Rightarrow y+4=12\Rightarrow x=8andy=8$

$\therefore$ Co-ordinates of the third vertex are (8, 8) Area of the $\Delta ABC$
$=\frac{1}{2}\left[\right(x_1{y}_2+x_2{y}_3+x_3{y}_1)-(x_1{y}_3+x_2{y}_1+x_3{y}_2\left)\right]=\frac{1}{2}\left[\right(8\times (-3)+4\times 7-9\times 8)-(8\times 7+4\times 8+(-9)\times (-3)\left)\right]=\frac{1}{2}[-68-115]=\frac{1}{2}\left[183\right]=\frac{183}{2}sq.units$