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Dallasb22

Find the area of an equilateral triangle (regular 3-gon) with the given measurement. 3-inch radius A = sq. in.

  • one year ago
  • one year ago

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  1. Dallasb22
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    Area= 1/2ap

    • one year ago
  2. Dallasb22
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    I draw an auxiliary triangle off of the radius, the radius being the hypotenuse

    • one year ago
  3. tcarroll010
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    By 3 inch radius, do you mean that it is inscribed in a circle of radius 3 inches?

    • one year ago
  4. Dallasb22
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    |dw:1348254489061:dw|

    • one year ago
  5. Dallasb22
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    The apothem is half of the radius

    • one year ago
  6. tcarroll010
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    |dw:1348254528658:dw| So it looks like both are drawings are similar?

    • one year ago
  7. Dallasb22
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    So its 1.5

    • one year ago
  8. Dallasb22
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    I multiply 1.5 times √3 to get the long side

    • one year ago
  9. Dallasb22
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    Than i multiply that answer "3√3" by 2 to get the side of the big triangle

    • one year ago
  10. Dallasb22
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    I multiply that by 3 to get the perimeter. So i have the perimeter and the apothem.

    • one year ago
  11. tcarroll010
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    Yes, and to make matters easier, the angle and segment bisectors will meet 2/3 of the way to the other side, making the height = 1.5 x apothem.

    • one year ago
  12. Dallasb22
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    The answer i got the first time is: 6 3/4√3 but they told me its wrong.

    • one year ago
  13. tcarroll010
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    height = 1.5 x apothem = 1.5 x (r/2) = 3r/4.

    • one year ago
  14. Dallasb22
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    I don't need the height to find the area though.

    • one year ago
  15. Dallasb22
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    All i need is the apothem and the perimeter

    • one year ago
  16. tcarroll010
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    You're right, you don't, but you can take advantage of some easy trigonometry to get the answer easier.

    • one year ago
  17. Dallasb22
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    Ok, well i'm only in Geometry.. So i'm not suppose to be doing trig

    • one year ago
  18. Dallasb22
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    *Well i don't know any

    • one year ago
  19. tcarroll010
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    Ok, then I'll stop with that approach.

    • one year ago
  20. Dallasb22
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    Lol, if it's fairly easy, I don't mind learning it

    • one year ago
  21. Dallasb22
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    It must be easier than the book's method because everyone I talk to on here starts using trig

    • one year ago
  22. tcarroll010
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    No, that's ok. I can do it the other way. That's the way you're supposed to be doing it and it will blow others away.

    • one year ago
  23. tcarroll010
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    I'll explain the methodology and you can do the work. It won't be that hard, really.

    • one year ago
  24. Dallasb22
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    I got most of my questions right, but this one i got wrong and i'm not sure why.

    • one year ago
  25. tcarroll010
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    We'll use apothem and perimeter and derive some cool measurements from them.

    • one year ago
  26. Dallasb22
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    Groovy

    • one year ago
  27. tcarroll010
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    We can use one thing I said above, that height = 3r/4 because that does come from the apothem to radius relationship and the fact that angle and side bisectors meet in the middle and make that height 3r/4. Stop and draw yourself that on paper to convince yourself. Because that's going to be key. And grooviness is good here!

    • one year ago
  28. Dallasb22
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    r= radius?

    • one year ago
  29. Dallasb22
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    9/4 is the height?

    • one year ago
  30. tcarroll010
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    yes.

    • one year ago
  31. tcarroll010
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    Now, here's the trick, and it's really cool conceptually, but a little hard to explain. maybe I can draw a picture.

    • one year ago
  32. Dallasb22
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    Ok.

    • one year ago
  33. tcarroll010
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    |dw:1348255695035:dw|

    • one year ago
  34. psi9epsilon
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    For an equilateral triangle circumcentre, incentre , centroid are all at the same point use this property to solve the problem

    • one year ago
  35. tcarroll010
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    I forgot to put a letter at the top, call it c and call the, here, I'll draw again

    • one year ago
  36. psi9epsilon
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    @tcarroll010 , has given you the correct diagram, just name the angles and measures of sides

    • one year ago
  37. tcarroll010
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    |dw:1348255924927:dw| There.

    • one year ago
  38. tcarroll010
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    We know ac = 9/4. We know oa is 3/4. We know ob is 3/4, so we can get ab and you can do 1/2 of h x b for oab. Then double for the whole triangle.

    • one year ago
  39. psi9epsilon
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    |dw:1348255963318:dw|

    • one year ago
  40. tcarroll010
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    Sorry, I did ob wrong, but you can do it. It's 2/3 of oa.

    • one year ago
  41. tcarroll010
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    ob is 2/3 of ac.

    • one year ago
  42. Dallasb22
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    Ob is 2/3 of 3/4?

    • one year ago
  43. tcarroll010
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    Sorry, I'm trying to do 3 things at once, but it's right there in the diagram and just use pythagorean on the lengths.

    • one year ago
  44. Dallasb22
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    I don't follow.

    • one year ago
  45. tcarroll010
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    2/3 of 9/4 so its 3/2

    • one year ago
  46. Dallasb22
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    1.5

    • one year ago
  47. tcarroll010
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    ob is 3/2 oa is 3/4, so you can get ab.

    • one year ago
  48. tcarroll010
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    ab will be your base. ac your height. Then you can easily get area of triangle.

    • one year ago
  49. Dallasb22
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    Ab being 1.5√3

    • one year ago
  50. Dallasb22
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    ac: 3

    • one year ago
  51. tcarroll010
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    ac is 9/2 or (4 and 1/2)

    • one year ago
  52. tcarroll010
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    oa is apothem which is 1/2 of r, so add 3 (which is oc) to 1 and 1/2 which is oa. so ac is 9/2

    • one year ago
  53. Dallasb22
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    Ahhh

    • one year ago
  54. tcarroll010
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    ob = oc = 3. So, you have a right triangle where you can figure out ab. ab^2 + oa^2 = ob^2

    • one year ago
  55. tcarroll010
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    Once you get ab, it's a piece of cake.

    • one year ago
  56. tcarroll010
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    Are you all set now?

    • one year ago
  57. tcarroll010
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    I gotta go. I hope you are all set.

    • one year ago
  58. Dallasb22
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    No.. Lol, ab^2 + 3/4^2 =3^2?

    • one year ago
  59. tcarroll010
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    I just got back to my computer after a half hour. You don't happen to still be there are you?

    • one year ago
  60. Dallasb22
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    Sadly, i am.

    • one year ago
  61. tcarroll010
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    My condolences. I'm thinking that you're not liking this problem anymore. I'm going to try to simplify it all in one post and you can just work from info in that and the diagram.

    • one year ago
  62. Dallasb22
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    OK

    • one year ago
  63. tcarroll010
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    oa is apothem and = r/2. oc=ob=r. ac=oc+oa. oa^2 + ab^2 = ob^2. Triangle area = ab x ac. That's all there is to it. My guess is that you're having trouble working with oa^2 + ab^2 = ob^2. So, for that ab: ab = sqrt(ob^2 - oa^2).

    • one year ago
  64. Dallasb22
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    oa= 3/2 right?

    • one year ago
  65. tcarroll010
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    yes

    • one year ago
  66. tcarroll010
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    Are you having trouble with the square root and getting ab? Is that the hangup?

    • one year ago
  67. Dallasb22
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    I don't think so, let me work out your equation above ^^

    • one year ago
  68. Dallasb22
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    OK, so i got: √6.75 So √6.75 times 4.5 = area? √6.75 = ab 4.5 = ac

    • one year ago
  69. tcarroll010
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    yes, you're getting it, you really are. It might be a little better to think of that ab length, which is sqrt(6.75), as(3 x sqrt(3)) / 2. But either way works. Whatever is easier for you.

    • one year ago
  70. tcarroll010
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    Because when you're done, and you really almost are, enough that I can give you the answer, the area is (27 x sqrt(3)) / 4.

    • one year ago
  71. Dallasb22
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    OKay.

    • one year ago
  72. tcarroll010
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    You should be able to get to that last number with what should be your upcoming and last step.

    • one year ago
  73. Dallasb22
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    So... 27√3 ------ 4 ???

    • one year ago
  74. Dallasb22
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    \[27√3\div4\]

    • one year ago
  75. tcarroll010
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    yes, but more importantly, do you see the flow of the steps? That's the important thing.

    • one year ago
  76. Dallasb22
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    I can't carry that out?

    • one year ago
  77. tcarroll010
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    You can put it in decimal from or any other representation if you want. Sqrt(3) is irrational, so any decimal representation will truncate digits, but that's ok.

    • one year ago
  78. tcarroll010
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    Decimal-wise, it can be truncated to 11.69134.

    • one year ago
  79. Dallasb22
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    Oh wait sorry, 27√3/4 is the area!! Okay

    • one year ago
  80. tcarroll010
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    yes, and area is ac x ab.

    • one year ago
  81. Dallasb22
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    I put the answer as: 27/4√3

    • one year ago
  82. Dallasb22
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    I'll let you know if they accept it in 5 minutes.

    • one year ago
  83. tcarroll010
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    Not 27/4√3. It's \[27\sqrt{3}/4\]

    • one year ago
  84. tcarroll010
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    The order of those multiplicands is important.

    • one year ago
  85. Dallasb22
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    Ok

    • one year ago
  86. Dallasb22
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    Correct Score: 100 Thank you!!

    • one year ago
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