## Dallasb22 2 years ago Find the area of an equilateral triangle (regular 3-gon) with the given measurement. 3-inch radius A = sq. in.

1. Dallasb22

Area= 1/2ap

2. Dallasb22

I draw an auxiliary triangle off of the radius, the radius being the hypotenuse

3. tcarroll010

By 3 inch radius, do you mean that it is inscribed in a circle of radius 3 inches?

4. Dallasb22

|dw:1348254489061:dw|

5. Dallasb22

The apothem is half of the radius

6. tcarroll010

|dw:1348254528658:dw| So it looks like both are drawings are similar?

7. Dallasb22

So its 1.5

8. Dallasb22

I multiply 1.5 times √3 to get the long side

9. Dallasb22

Than i multiply that answer "3√3" by 2 to get the side of the big triangle

10. Dallasb22

I multiply that by 3 to get the perimeter. So i have the perimeter and the apothem.

11. tcarroll010

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

The answer i got the first time is: 6 3/4√3 but they told me its wrong.

13. tcarroll010

height = 1.5 x apothem = 1.5 x (r/2) = 3r/4.

14. Dallasb22

I don't need the height to find the area though.

15. Dallasb22

All i need is the apothem and the perimeter

16. tcarroll010

You're right, you don't, but you can take advantage of some easy trigonometry to get the answer easier.

17. Dallasb22

Ok, well i'm only in Geometry.. So i'm not suppose to be doing trig

18. Dallasb22

*Well i don't know any

19. tcarroll010

Ok, then I'll stop with that approach.

20. Dallasb22

Lol, if it's fairly easy, I don't mind learning it

21. Dallasb22

It must be easier than the book's method because everyone I talk to on here starts using trig

22. tcarroll010

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

I'll explain the methodology and you can do the work. It won't be that hard, really.

24. Dallasb22

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

25. tcarroll010

We'll use apothem and perimeter and derive some cool measurements from them.

26. Dallasb22

Groovy

27. tcarroll010

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

29. Dallasb22

9/4 is the height?

30. tcarroll010

yes.

31. tcarroll010

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

Ok.

33. tcarroll010

|dw:1348255695035:dw|

34. psi9epsilon

For an equilateral triangle circumcentre, incentre , centroid are all at the same point use this property to solve the problem

35. tcarroll010

I forgot to put a letter at the top, call it c and call the, here, I'll draw again

36. psi9epsilon

@tcarroll010 , has given you the correct diagram, just name the angles and measures of sides

37. tcarroll010

|dw:1348255924927:dw| There.

38. tcarroll010

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

|dw:1348255963318:dw|

40. tcarroll010

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

41. tcarroll010

ob is 2/3 of ac.

42. Dallasb22

Ob is 2/3 of 3/4?

43. tcarroll010

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

I don't follow.

45. tcarroll010

2/3 of 9/4 so its 3/2

46. Dallasb22

1.5

47. tcarroll010

ob is 3/2 oa is 3/4, so you can get ab.

48. tcarroll010

ab will be your base. ac your height. Then you can easily get area of triangle.

49. Dallasb22

Ab being 1.5√3

50. Dallasb22

ac: 3

51. tcarroll010

ac is 9/2 or (4 and 1/2)

52. tcarroll010

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

Ahhh

54. tcarroll010

ob = oc = 3. So, you have a right triangle where you can figure out ab. ab^2 + oa^2 = ob^2

55. tcarroll010

Once you get ab, it's a piece of cake.

56. tcarroll010

Are you all set now?

57. tcarroll010

I gotta go. I hope you are all set.

58. Dallasb22

No.. Lol, ab^2 + 3/4^2 =3^2?

59. tcarroll010

I just got back to my computer after a half hour. You don't happen to still be there are you?

60. Dallasb22

61. tcarroll010

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

OK

63. tcarroll010

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

oa= 3/2 right?

65. tcarroll010

yes

66. tcarroll010

Are you having trouble with the square root and getting ab? Is that the hangup?

67. Dallasb22

I don't think so, let me work out your equation above ^^

68. Dallasb22

OK, so i got: √6.75 So √6.75 times 4.5 = area? √6.75 = ab 4.5 = ac

69. tcarroll010

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

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

OKay.

72. tcarroll010

You should be able to get to that last number with what should be your upcoming and last step.

73. Dallasb22

So... 27√3 ------ 4 ???

74. Dallasb22

$27√3\div4$

75. tcarroll010

yes, but more importantly, do you see the flow of the steps? That's the important thing.

76. Dallasb22

I can't carry that out?

77. tcarroll010

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

Decimal-wise, it can be truncated to 11.69134.

79. Dallasb22

Oh wait sorry, 27√3/4 is the area!! Okay

80. tcarroll010

yes, and area is ac x ab.

81. Dallasb22

I put the answer as: 27/4√3

82. Dallasb22

I'll let you know if they accept it in 5 minutes.

83. tcarroll010

Not 27/4√3. It's $27\sqrt{3}/4$

84. tcarroll010

The order of those multiplicands is important.

85. Dallasb22

Ok

86. Dallasb22

Correct Score: 100 Thank you!!