Question

Find the areas of the triangles the whose coordinates of the points are respectively.

(a, b + c), (a, b-c) and (-a, c)

Answer 1

Area of triangle having angular points
$(x_1,y_1)$ $(x_2,y_2)$ $(x_3,y_3)$ is given by the formula

$△=\frac{1}{2}|x_1{y}_2+x_2{y}_3+x_3{y}_1-x_2{y}_1-x_3{y}_2-x_1{y}_3|$

or $△=\frac{1}{2}\left|x_1\right(y_2-y_3)+x_2(y_3-y_1)+x_3(y_1-y_2\left)\right|$

Points given are $(a,b+c),(a,b-c)$ and $(-a,c)$

Then, $Area=△=\frac{1}{2}\left|a\right(b-c-c)+a(c-b-c)+(-a\left)\right(b+c-(b-c)\left)\right|$

$\Rightarrow Area=△=\frac{1}{2}|ab-2ac-ab-2ac|$

$\Rightarrow Area=△=\frac{1}{2}|-4ac|$

$\Rightarrow Area=△=2ac$