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Question: Let A<sub>0</sub> A<sub>1</sub> A<sub>2</sub> A<sub>3</sub> A<sub>4</sub> A<sub>5</sub>be a regular ...

Let A0 A1 A2 A3 A4 A5be a regular hexagon inscribed in a unit circle with centre at the origin. Then the product of the lengths of the line segments A0 A1, A0 A2 and A0 A4 is-

A

¾

B

333 \sqrt { 3 }

C

3

D

333 \sqrt { 3 }/2

Answer

3

Explanation

Solution

Let O be the centre of the circle of unit radius and the coordinates of A0 be (1, 0).

Since each side of the regular hexagon makes an angle of 600 at the centre O.

Coordinates of A1 are (cos 600, sin 600) =(12,32)\left( \frac { 1 } { 2 } , \frac { \sqrt { 3 } } { 2 } \right)

A2 are (cos 120°, sin 1200) = (−12,32)\left( - \frac { 1 } { 2 } , \frac { \sqrt { 3 } } { 2 } \right)

A3 are (–1, 0)

A4 are (12,−32)\left( \frac { 1 } { 2 } , - \frac { \sqrt { 3 } } { 2 } \right)

Now A0 A1 = (1−12)2+(32)2\sqrt { \left( 1 - \frac { 1 } { 2 } \right) ^ { 2 } + \left( \frac { \sqrt { 3 } } { 2 } \right) ^ { 2 } }= 14+34\sqrt { \frac { 1 } { 4 } + \frac { 3 } { 4 } } = 1

A0 A2 = 3\sqrt { 3 } = A0 A4

So that (A0 A1) (A0 A2) (A0 A4) = 3.