Dallasb22
Find the area of an equilateral triangle (regular 3-gon) with the given measurement.
3-inch radius
A = sq. in.
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Dallasb22
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Area= 1/2ap
Dallasb22
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I draw an auxiliary triangle off of the radius, the radius being the hypotenuse
tcarroll010
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By 3 inch radius, do you mean that it is inscribed in a circle of radius 3 inches?
Dallasb22
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|dw:1348254489061:dw|
Dallasb22
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The apothem is half of the radius
tcarroll010
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|dw:1348254528658:dw| So it looks like both are drawings are similar?
Dallasb22
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So its 1.5
Dallasb22
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I multiply 1.5 times √3 to get the long side
Dallasb22
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Than i multiply that answer "3√3" by 2 to get the side of the big triangle
Dallasb22
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I multiply that by 3 to get the perimeter. So i have the perimeter and the apothem.
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.
Dallasb22
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The answer i got the first time is: 6 3/4√3 but they told me its wrong.
tcarroll010
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height = 1.5 x apothem = 1.5 x (r/2) = 3r/4.
Dallasb22
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I don't need the height to find the area though.
Dallasb22
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All i need is the apothem and the perimeter
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.
Dallasb22
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Ok, well i'm only in Geometry.. So i'm not suppose to be doing trig
Dallasb22
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*Well i don't know any
tcarroll010
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Ok, then I'll stop with that approach.
Dallasb22
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Lol, if it's fairly easy, I don't mind learning it
Dallasb22
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It must be easier than the book's method because everyone I talk to on here starts using trig
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.
tcarroll010
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I'll explain the methodology and you can do the work. It won't be that hard, really.
Dallasb22
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I got most of my questions right, but this one i got wrong and i'm not sure why.
tcarroll010
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We'll use apothem and perimeter and derive some cool measurements from them.
Dallasb22
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Groovy
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!
Dallasb22
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r= radius?
Dallasb22
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9/4 is the height?
tcarroll010
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yes.
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.
Dallasb22
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Ok.
tcarroll010
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|dw:1348255695035:dw|
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
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
psi9epsilon
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@tcarroll010 , has given you the correct diagram, just name the angles and measures of sides
tcarroll010
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|dw:1348255924927:dw| There.
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.
psi9epsilon
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|dw:1348255963318:dw|
tcarroll010
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Sorry, I did ob wrong, but you can do it. It's 2/3 of oa.
tcarroll010
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ob is 2/3 of ac.
Dallasb22
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Ob is 2/3 of 3/4?
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.
Dallasb22
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I don't follow.
tcarroll010
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2/3 of 9/4 so its 3/2
Dallasb22
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1.5
tcarroll010
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ob is 3/2 oa is 3/4, so you can get ab.
tcarroll010
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ab will be your base. ac your height. Then you can easily get area of triangle.
Dallasb22
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Ab being 1.5√3
Dallasb22
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ac: 3
tcarroll010
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ac is 9/2 or (4 and 1/2)
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
Dallasb22
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Ahhh
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
tcarroll010
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Once you get ab, it's a piece of cake.
tcarroll010
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Are you all set now?
tcarroll010
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I gotta go. I hope you are all set.
Dallasb22
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No.. Lol, ab^2 + 3/4^2 =3^2?
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?
Dallasb22
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Sadly, i am.
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.
Dallasb22
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OK
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).
Dallasb22
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oa= 3/2 right?
tcarroll010
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yes
tcarroll010
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Are you having trouble with the square root and getting ab? Is that the hangup?
Dallasb22
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I don't think so, let me work out your equation above ^^
Dallasb22
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OK, so i got: √6.75
So √6.75 times 4.5 = area?
√6.75 = ab
4.5 = ac
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.
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.
Dallasb22
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OKay.
tcarroll010
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You should be able to get to that last number with what should be your upcoming and last step.
Dallasb22
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So...
27√3
------
4
???
Dallasb22
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\[27√3\div4\]
tcarroll010
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yes, but more importantly, do you see the flow of the steps? That's the important thing.
Dallasb22
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I can't carry that out?
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.
tcarroll010
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Decimal-wise, it can be truncated to 11.69134.
Dallasb22
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Oh wait sorry,
27√3/4 is the area!!
Okay
tcarroll010
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yes, and area is ac x ab.
Dallasb22
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I put the answer as: 27/4√3
Dallasb22
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I'll let you know if they accept it in 5 minutes.
tcarroll010
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Not 27/4√3. It's \[27\sqrt{3}/4\]
tcarroll010
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The order of those multiplicands is important.
Dallasb22
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Ok
Dallasb22
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Correct Score: 100
Thank you!!