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Dallasb22

  • 2 years ago

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

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  1. Dallasb22
    • 2 years ago
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    Area= 1/2ap

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

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

  4. Dallasb22
    • 2 years ago
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    |dw:1348254489061:dw|

  5. Dallasb22
    • 2 years ago
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    The apothem is half of the radius

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

  7. Dallasb22
    • 2 years ago
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    So its 1.5

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

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

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

  11. tcarroll010
    • 2 years ago
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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.

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

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

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

  15. Dallasb22
    • 2 years ago
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    All i need is the apothem and the perimeter

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

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

  18. Dallasb22
    • 2 years ago
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    *Well i don't know any

  19. tcarroll010
    • 2 years ago
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    Ok, then I'll stop with that approach.

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

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

  22. tcarroll010
    • 2 years ago
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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.

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

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

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

  26. Dallasb22
    • 2 years ago
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    Groovy

  27. tcarroll010
    • 2 years ago
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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!

  28. Dallasb22
    • 2 years ago
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    r= radius?

  29. Dallasb22
    • 2 years ago
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    9/4 is the height?

  30. tcarroll010
    • 2 years ago
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    yes.

  31. tcarroll010
    • 2 years ago
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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.

  32. Dallasb22
    • 2 years ago
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    Ok.

  33. tcarroll010
    • 2 years ago
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    |dw:1348255695035:dw|

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

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

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

  37. tcarroll010
    • 2 years ago
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    |dw:1348255924927:dw| There.

  38. tcarroll010
    • 2 years ago
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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.

  39. psi9epsilon
    • 2 years ago
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    |dw:1348255963318:dw|

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

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

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

  43. tcarroll010
    • 2 years ago
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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.

  44. Dallasb22
    • 2 years ago
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    I don't follow.

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

  46. Dallasb22
    • 2 years ago
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    1.5

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

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

  49. Dallasb22
    • 2 years ago
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    Ab being 1.5√3

  50. Dallasb22
    • 2 years ago
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    ac: 3

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

  52. tcarroll010
    • 2 years ago
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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

  53. Dallasb22
    • 2 years ago
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    Ahhh

  54. tcarroll010
    • 2 years ago
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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

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

  56. tcarroll010
    • 2 years ago
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    Are you all set now?

  57. tcarroll010
    • 2 years ago
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    I gotta go. I hope you are all set.

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

  59. tcarroll010
    • 2 years ago
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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?

  60. Dallasb22
    • 2 years ago
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    Sadly, i am.

  61. tcarroll010
    • 2 years ago
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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.

  62. Dallasb22
    • 2 years ago
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    OK

  63. tcarroll010
    • 2 years ago
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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).

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

  65. tcarroll010
    • 2 years ago
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    yes

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

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

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

  69. tcarroll010
    • 2 years ago
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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.

  70. tcarroll010
    • 2 years ago
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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.

  71. Dallasb22
    • 2 years ago
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    OKay.

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

  73. Dallasb22
    • 2 years ago
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    So... 27√3 ------ 4 ???

  74. Dallasb22
    • 2 years ago
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    \[27√3\div4\]

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

  76. Dallasb22
    • 2 years ago
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    I can't carry that out?

  77. tcarroll010
    • 2 years ago
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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.

  78. tcarroll010
    • 2 years ago
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    Decimal-wise, it can be truncated to 11.69134.

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

  80. tcarroll010
    • 2 years ago
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    yes, and area is ac x ab.

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

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

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

  84. tcarroll010
    • 2 years ago
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    The order of those multiplicands is important.

  85. Dallasb22
    • 2 years ago
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    Ok

  86. Dallasb22
    • 2 years ago
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    Correct Score: 100 Thank you!!

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