Question

# Let A_{0} A_{1} A_{2} A_{3} A_{4} A_{5} be a regular hexagon inscribed in a circle of unit radius. Then the product of the lengths of the line segments A_{0}A_{1}, A_{0}A_{2}, and A_{0}A_{4} is -

- 3/4
- 3

Hint:

### Here A0 A1 A2 A3 A4 A5 be regular hexagon inscribe in circle of unit radius. Here we have to find that the product of length of the A0A1, A0 A2 and A0 A4. Now three circle is given inside. Find the length of all these sides and multiply them.

## The correct answer is: 3

### Here we have to find the product of A0A1, A0 A2 and A0 A4.

Firstly, we have a Hexagon inscribed in a circle, let center of circle be O.

In given figure,

Here OA0=1

Then,

OA1=OA2=OA3=OA4=OA5=1

Since, it is regular hexagon

therefore,

All sides are equal

And,

Each side of hexagon makes an angle 60∘ at the Centre O of the circle coordinates of A1, A2, A4, A5 are

(cos60∘, sin60∘), (cos120∘, sin120∘), (−cos60∘, sin60∘), (−cos120∘, −sin120∘) respectively.

A1= (, )

A2= (, )

A4= (, )

A5= (, )

And

A3= (−1,0) and A0= (1,0) (given circle is of radius.)

Now,

By distance formula,

Length of A0A1,

(A0A1)^{2} = ( − 1)^{2} + ( −0)^{2}

So,

A0A1=1

Now,

Length of A0A2,

Similarly,

A0A2 =

And,

Length of A0A4,

Similarly,

A0A4 =

And,

the product of lengths of the line segments A0A1, A0A2 and A0A4 is

=1 × ×

= 3

Therefore , the product of length of A0A1, A0A2 and A0A4 is 3.

The correct answer is 3.

In this question we have to find the product of length of A0A1, A0A2 and A0A4. Firstly, find the length of these sides. Each angle of the regular hexagon is 120 and each side are equal.

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