Geometry and Ratio problem!!! Help!!!

- anonymous

Geometry and Ratio problem!!! Help!!!

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- anonymous

The figure ABCDEF below is a regular hexagon, and point P lies on side AB, with AP=3 cm and PB=1cm. Line PE meets AD at Q. What is the ratio of the area of quadrilateral AQEF to the area of hexagon ABCDEF? Express your answer as a common fraction.

- anonymous

|dw:1327787358619:dw|

- Hero

All you do is find the areas and put them in fraction form

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- anonymous

how do I find the area of AQEF?

- Hero

It's not that difficult

- anonymous

Can you give me some hints? Please.

- Hero

|dw:1327787517063:dw|

- Hero

There's one hint

- Hero

Ready for the next hint?

- anonymous

yes

- Hero

Ready for the next one?

- anonymous

Wait, so all the side lengths are 3? Isn't it 4?

- Hero

oh okay, I didn't read that in-depth. This is different

- anonymous

ok

- Hero

|dw:1327787748238:dw|

- anonymous

Yes

- Hero

You have to find the area of the hexagon first. There's no way to avoid that

- Hero

Good luck with this

- anonymous

Area of the hexagon = |dw:1327788115227:dw|

- Hero

I don't have a solid approach to this. Sorry bud

- Hero

I can take a look at it later maybe.

- anonymous

Oh, ok. Thanks for trying though.

- Hero

AD splits the hex into two halves. That might help.

- anonymous

ok, I will keep trying.

- Hero

|dw:1327788413768:dw|

- Hero

|dw:1327788548241:dw|

- anonymous

Area of hexagon equals 24sqrt{3}, so ABCD equals 12sqrt{3} and ADEF equals 12 sqrt{3} as well.

- anonymous

find the heigh of AE, calcuate the area of triangle APE and AFE...then you have an answer
i guess

- anonymous

the height of AE is 4sqrt{3}, APE is 6sqrt{3}.
But how do I find AFE and AQP then?

- anonymous

|dw:1327789175987:dw|

- anonymous

What about AQE?

- anonymous

I made an mistake, AEF should be 4sqrt{3}.

- Hero

asnaseer, moneybird, and mertsj should be able to figure this out.

- asnaseer

I think matematika and yociyoci are doing well so far...

- anonymous

im not sure....but isn't there formula for calulating triangle if you know one side and angels?maybe with help of this you can find out area of this one

- anonymous

But I don't know that formula, is there another approach?

- anonymous

You can find the height of triangle EDQ

- anonymous

triangle EDQ and triangle PAQ are similar

- anonymous

so the height of EDQ is|dw:1327790090893:dw| ?

- anonymous

AD splits the hexagon in halves so quadrilateral AQEF = quadrilateral AFED - triangle FDQ

- asnaseer

you mean - EDQ moneybird

- anonymous

is triangle EDQ eqauls to 32sqrt{3}/7 ?

- anonymous

ya
|dw:1327790174473:dw|
From midpoint of ED to midpoint of AB is 4 sqrt(3)

- anonymous

now use ratio to find the height of triangle EDQ and triangle APQ

- anonymous

\[\frac{4}{h} = \frac{3}{4\sqrt{3}-h}\]

- anonymous

I don't quite understand what you just did. Can you clarify, please?

- anonymous

Which part?

- anonymous

the diagram

- anonymous

when you cut the hexagon into six equal pieces, you take out one of the piece, which is a triangle

- anonymous

ok

- anonymous

|dw:1327790925432:dw|so the area of the hexagon is 24 sqrt(3)
ratio is

- anonymous

ok, i understand.
thank you very much, guys!!!!
I appreciated!!!

- asnaseer

very well explained moneybird

- anonymous

ty

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