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In the figure above, ABCDEF is a regular hexagon with

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anmolmakkarz17
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In the figure above, ABCDEF is a regular hexagon with [#permalink] New post   01 Nov 2019, 10:42
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In the figure above, ABCDEF is a regular hexagon with vertices A and C on the circle centered at point B. Is the perimeter of the figure greater than 40 ?

(1) The overlapping region between circle B and hexagon ABCDEF has an area of 12π.
(2) The area of the figure is 24\(\pi\) +54 \(\sqrt{3}\)

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Re: In the figure above, ABCDEF is a regular hexagon with [#permalink] New post   01 Nov 2019, 11:27
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Before looking at the statements, there are few key points that we need to take care of..

First- this is a regular hexagon. Thus all the sides are equal in length and each internal angle is 120° { ((n-2)*180)/N, where N= 6}
Second- radius= side of hexagon..

Therefore to find whether perimeter is greater than 40, we just need to find the radius or length of each side of hexagon..

Statement 1- overlapping area is 12π.. we can get the radius and hence find the perimeter of entire figure. Sufficient

Statement 2- area of the figure is given..
=> Area of circle - area of sector + area of hexagon = 24π + 54√3..

We can find out the radius "r" and hence the perimeter. Sufficient.

Each statement is sufficient on its own.
D...


P.s - I didn't calculate the the values because we actually don't need to. Since we know that each statement will give a value of "r", a unique value, we will have a unique answer...


However for conceptual basis-

Statement 1 - 120/360*π*r^2
=> 1/3*π*r^2 = 12π
=> r^2 = 36
=> r = 6

Statement 2 - area of Figure = 24π + 54√3
=> π*r^2 - (1/3)*π*r^2 + (√3/4)*r^2*6
=> (2/3)π*r^2 + (3√3/2)*r^2
=> (2/3)π*r^2 = 24π
=> r^2 = 24*(3/2) => r^2 = 36
=> r = 6.

Hope it helps!

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Re: In the figure above, ABCDEF is a regular hexagon with [#permalink] New post   01 Nov 2019, 11:43
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The overlapping area is \(\frac{(πr^2)}{3}\). We will know the perimeter of the figure once we have the value of r (r=the radius of the circle and also the side of the regular hexagon)

St 1) We have the value of \(\frac{(πr^2)}{3}\). We can find r. SUFFICIENT
St 2) We have the total area of the figure, which is (\(\frac{2}{3}\)*\((πr^2)\)) +\(\frac{6\sqrt{3}}{4}*r^2\). We get the value of r from here too. SUFFICIENT

Hence D

Some formulas or value used in above calculation, that may not look so obvious to some
Area of a regular hexagon = \(\frac{6\sqrt{3}}{4}*r^2\)
Each angle of a hexagon is 120 degree, hence the sector area= \(\frac{120}{360} * πr^2\)= \(\frac{(πr^2)}{3}\)
Are of the the remaining sector = \((πr^2)\)- \(\frac{(πr^2)}{3}\)= \(\frac{2(πr^2)}{3}\)
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Re: In the figure above, ABCDEF is a regular hexagon with [#permalink] New post   02 Nov 2019, 06:14
In (1), I did take the time to find out that r = 6. Then it should be sufficient.
For (2), I learnt to be smart to decide that as long as I can find r (whatever it is), it provides sufficient data to answer the question.

The moral of the story is that: there is NO need to answer the question. If we know we have enough info to solve the question, that is sufficient.
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Re: In the figure above, ABCDEF is a regular hexagon with [#permalink] New post   03 Nov 2019, 00:03
Don't get bogged down by big numbers and complicated calculations. The general prep strategy for Data Sufficiency (DS) questions is that there is no need to solve the equations or do the math.

The radius of the circle is equal to the side of the hexagon. Let the radius of the circle be r; therefore the side of the hexagon is also r.

St1) The overlapping region between circle B and hexagon ABCDEF has an area of 12π. i.e. the area of the sector is given. We can find the angle of a regular hexagon and that will also be the angle of the sector. From this equation, we can find the value of the r and therefore answer the question "Is the perimeter of the figure greater than 40 ?" - sufficient

St2) The area of the figure is given. Which is same as [Area of the circle - the area of sector + area of the hexagon]
Again we can find the value of the r and answer the question "Is the perimeter of the figure greater than 40 ?" - sufficient

Option D is the correct answer.
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Re: In the figure above, ABCDEF is a regular hexagon with [#permalink] New post   15 Jul 2020, 23:55
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anmolmakkarz17 wrote:
In the figure above, ABCDEF is a regular hexagon with vertices A and C on the circle centered at point B. Is the perimeter of the figure greater than 40 ?

1) The overlapping region between circle B and hexagon ABCDEF has an area of 12π.
2) The area of the figure is 24\(\pi\) +54 \(\sqrt{3}\)


B is the centre of the circle and the vertex of a regular hexagon. So angle ABC will be 120 degrees.
This means sector ABC will be 1/3rd the area of the triangle. The radius BC (or BA) is also the side of the hexagon. If we know its measure, we will know the area of the circle as well as the area of the hexagon.
Also, the regular hexagon is made up of 6 equal equilateral triangles. So if we know the area of the hexagon, we know the area of each triangle and the measure of each side of the hexagon.

Ques: Is the perimeter of the figure greater than 40 ?
If we know the measure of the side of the figure, we will know whether its perimeter will be greater than 40. So what we need is the measure of the side of the hexagon.

1) The overlapping region between circle B and hexagon ABCDEF has an area of 12π.

This tells us that area of the sector is 12π. This is 1/3rd the area of the circle so we can find the area of the circle. Then we can find the radius of the circle which is same as side of the hexagon. Sufficient alone.

2) The area of the figure is 24
This is the area of the figure which is 6 times the area of each equilateral triangle. So we can find the area of each equilateral triangle which will give us the measure of each side of hexagon.
Sufficient alone

Answer (D)
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Re: In the figure above, ABCDEF is a regular hexagon with [#permalink]
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