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H1 is a regular hexagon circumscribing a...

`H_1` is a regular hexagon circumscribing a circle. `H_2` is a regular hexagon inscribed in the circle. Find the ratio of areas of `H_1` and `H_2`

A

`4:3`

B

`2:1`

C

`3:1`

D

`3:2`

Text Solution

Verified by Experts

The correct Answer is:
A

Let `H_(1)` be PQRSTU and `H_(2)` be ABCDEF.
Let R be the radius of the circle.
` :. R=(sqrt(3)(PU))/(2)implies PU=(2R)/(sqrt(3))`
`AB=R ( :' "A hexagon inscribed in a circle must have its side equal to the radius of the circle")`
Area of `H_(1)=(3 sqrt(3))/(2)((2)/(sqrt(3))R)^(2)`
And area of `H_(2)=(3sqrt(3))/(2)(R)^(2)`
`( :' " Area of hexagon"=(3sqrt(3))/(2) xx ("Its side")^(2))`
Required ratio `=(((2)/(sqrt(3))R)^(2))/(R^(2))=(4)/(3)`.
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