Question

If the coordinates of the vertices of a triangle given in polar form are $(0,0),(3,\frac{\pi }{3})$ and $(3,\frac{2\pi }{3}),$ then the perimeter of the triangle is

Answer 1

Given points of the triangle are
$A=(0,0),B=(3,\frac{\pi }{3}),C=(3,\frac{2\pi }{3})$
Changing the coordinates in Cartesian form, we get
$A=(0,0)B=(3\cos \frac{\pi }{3},3\sin \frac{\pi }{3})=(\frac{3}{2},\frac{3\sqrt{3}}{2})C=(3\cos \frac{2\pi }{3},3\sin \frac{2\pi }{3})=(-\frac{3}{2},\frac{3\sqrt{3}}{2})$

Now, $AB=3,$ $BC=3,AC=3$
Therefore, it is an equilateral triangle with side length $=3$, so the perimeter $=9$