Question

The polar coordinates of the vertices of a triangle are $(0,0),$ $(3,\frac{\pi }{2})$ and $(3,\frac{\pi }{6})$. Then the triangle is

• A. right angled
• B. isosceles
• C. equilateral
• D. none of these

Answer 1

The correct option is C equilateral

Given $A(0,0),B(3,\frac{\pi }{2}),C(3,\frac{\pi }{6})$
$d=\sqrt{r_1^2+r_2^2-2r_1{r}_2\cos ({\theta }_1-{\theta }_2)}$
$\Rightarrow AB=\sqrt{0+9-0}=3$
$BC=\sqrt{9+9-18\cos \left(\frac{\pi }{3}\right)}$
$\Rightarrow BC=3$
$AC=\sqrt{0+9-0}=3$
$\Rightarrow AB=BC=AC$
Hence, the triangle is equilateral.