determine the ratio of the area of the outer regular pentagon to the area of the shaded inner region

Question

anonymous · Answer

attachment

kasim17 · Answer

1/10th

anonymous · Answer

which grade are you in?

anonymous · Answer

what you can do is assume the length of a side of the pentagon as 5, then bruteforce the solution

anonymous · Answer

11th

anonymous · Answer

idk what you said in the second comment means??

anonymous · Answer

but how? can you explain how you got it please?

anonymous · Answer

ok so.. first what is the angle of each corner of pentagon

anonymous · Answer

i dont know it dosent say anything about it ?

anonymous · Answer

a pentagon has 540 degree as inner angle, hence each corner has 540/5=108 degree

anonymous · Answer

ohhh :O im sorry i feel so stupid :/

anonymous · Answer

the final answer is 6.85

anonymous · Answer

tell me if you wanna know how to do it

anonymous · Answer

yes i do :O

anonymous · Answer

easier to explain here: https://docs.google.com/drawings/d/1Ez2TDLcr-tM2E6ye-ptQ5KAmy2sSIsTtmYVZ2VQrhJE/edit

anonymous · Answer

ok

anonymous · Answer

do you get it now...

anonymous · Answer

no -___________________________-

anonymous · Answer

Let me give it a try?

All you need to do to find the answer to this problem is find the ratio between the lengths of the sides of the pentagon.  The ratio of the areas of the pentagons is the ratio of the lengths, squared.  If this sounds complicated, think of a square.  For a square with a side length of four, the length is four times that of a square with a length of 1, and the area is 16 times that of a square with a length of 1.

Okay, now that that's out of the way, we can take a look at our pentagon:|dw:1357514712589:dw|

To find the ratio of s2 to s, we can find the following ratio:|dw:1357514938310:dw|

Because the pentagon is a regular pentagon, we already know the external angles (shown in the first picture). So, L1 = s * sin(36) = 0.588s.  L2 = s * sin (72) = 0.951s.

L1 : L1 + L2 = 0.588 : 0.588 + 0.951 = 0.382

So, the ratio of the sides of the two pentagons is 0.382.  This means that the area ratio squared is (0.382)^2 = 0.146

The ratio of the outer pentagon to the inner pentagon is 1 / 0.146, or 6.85.

I can see why this would be a difficult question.  If you have any questions, please ask!