## anonymous 4 years ago Geometry and Ratio problem!!! Help!!!

1. 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.

2. anonymous

|dw:1327787358619:dw|

3. Hero

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

4. anonymous

how do I find the area of AQEF?

5. Hero

It's not that difficult

6. anonymous

Can you give me some hints? Please.

7. Hero

|dw:1327787517063:dw|

8. Hero

There's one hint

9. Hero

10. anonymous

yes

11. Hero

12. anonymous

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

13. Hero

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

14. anonymous

ok

15. Hero

|dw:1327787748238:dw|

16. anonymous

Yes

17. Hero

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

18. Hero

Good luck with this

19. anonymous

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

20. Hero

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

21. Hero

I can take a look at it later maybe.

22. anonymous

Oh, ok. Thanks for trying though.

23. Hero

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

24. anonymous

ok, I will keep trying.

25. Hero

|dw:1327788413768:dw|

26. Hero

|dw:1327788548241:dw|

27. anonymous

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

28. anonymous

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

29. anonymous

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

30. anonymous

|dw:1327789175987:dw|

31. anonymous

32. anonymous

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

33. Hero

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

34. asnaseer

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

35. 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

36. anonymous

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

37. anonymous

You can find the height of triangle EDQ

38. anonymous

triangle EDQ and triangle PAQ are similar

39. anonymous

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

40. anonymous

41. asnaseer

you mean - EDQ moneybird

42. anonymous

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

43. anonymous

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

44. anonymous

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

45. anonymous

$\frac{4}{h} = \frac{3}{4\sqrt{3}-h}$

46. anonymous

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

47. anonymous

Which part?

48. anonymous

the diagram

49. anonymous

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

50. anonymous

ok

51. anonymous

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

52. anonymous

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

53. asnaseer

very well explained moneybird

54. anonymous

ty