Question

Find the areas of the triangles the coordinates of whose angular points are $(-3,-{30}^o),(5,{150}^o)$ and $(7,{210}^o)$

Answer 1

Given polar co-ordinates

(i) $(-3,-{30}^{\circ })$

Then, $r=-3$ and $\theta =-{30}^{\circ }$

$x=rcos\theta =(-3)cos(-{30}^{\circ })=\frac{-3\sqrt{3}}{2}$

$y=rsin\theta =(-3)sin(-{30}^{\circ })=\frac{3}{2}$

$A(\frac{-3\sqrt{3}}{2},\frac{3}{2})$

(ii) $(5,{150}^{\circ })$

Then, $r=5$ and $\theta ={150}^{\circ }$

$x=rcos\theta =\left(5\right)cos\left({150}^{\circ }\right)=\frac{-5\sqrt{3}}{2}$

$y=rsin\theta =\left(5\right)sin\left({150}^{\circ }\right)=\frac{5}{2}$

$B(\frac{-5\sqrt{3}}{2},\frac{5}{2})$

(iii) $(7,{210}^{\circ })$

Then, $r=7$ and $\theta ={210}^{\circ }$

$x=rcos\theta =\left(7\right)cos\left({210}^{\circ }\right)=\frac{-7\sqrt{3}}{2}$

$y=rsin\theta =\left(7\right)sin\left({210}^{\circ }\right)=\frac{-7}{2}$

$C(\frac{-7\sqrt{3}}{2},\frac{-7}{2})$

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

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

$A\left(△ABC\right)=\frac{1}{2}|\frac{-15\sqrt{3}}{4}+\frac{35\sqrt{3}}{4}+\frac{-21\sqrt{3}}{4}-(-\frac{15\sqrt{3}}{4})-(-\frac{35\sqrt{3}}{4})-\left(\frac{21\sqrt{3}}{4}\right)|=\frac{7\sqrt{3}}{2}$