Question

The polar coordinates of the vertices of a triangle are \[(0,{\mathbf{0}}),\;\left( {3,\dfrac{\pi }{2}} \right)\] and \[\left( {3,\dfrac{\pi }{6}} \right)\] . Then the triangle is
A.Right angled
B.Isosceles
C.Equilateral
D.None of these

Answer 1

Hint: To check which type of triangle is this whose polar coordinate is given. Generally triangles are differentiated by their side length. So we will find the side length of the triangle using the distance formula in between two polar coordinates. Such that if all the side has same length it will be an equilateral triangle, if two sides have the same length, triangle will be isosceles triangle, if the sum of square of any two side is equal to the square of the remaining side of the triangle then the triangle will be right angle triangle.

Complete step-by-step answer:
Given \[A\left( {0,0} \right),B\left( {3,\dfrac{\pi }{2}} \right),C\left( {3,\dfrac{\pi }{6}} \right)\]
we know the distance between two point in polar coordinate system
 \[d = \sqrt {{r_1}^2 + {r_2}^2 - 2{r_1}{r_2}cos\left( {{\theta _1} - {\theta _2}} \right)} \]
so the length of the side AB
 \[ \Rightarrow AB = \sqrt {0 + 9 - 0} = 3\]
similarly side BC
 \[BC = \sqrt {9 + 9 - 18cos\left( {\dfrac{\pi }{3}} \right)} \]
 \[ \Rightarrow BC = 3\]
side AC
 \[AC = \sqrt {0 + 9 - 0} = 3\]
Here we can see the side length of all the sides of the triangle is exactly same
 \[ \Rightarrow AB = BC = AC\]
Hence, the triangle is equilateral.
So, the correct answer is “Option C”.

Note: This problem can also be solved by converting all the vertices from the polar coordinate system to the Cartesian coordinate system initially then all the procedure is same like finding each side length and comparing all then answer the type of triangle.