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An equilateral triangle is inscribed in a circle. This

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An equilateral triangle is inscribed in a circle. This [#permalink] New post 11 May 2004, 02:47
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An equilateral triangle is inscribed in a circle. This circle is inscribed in another equilateral triangle. What is the ratio of the perimeter of the large triangle to that of the small one.
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Re: PS-33 [#permalink] New post 11 May 2004, 03:12
hallelujah1234 wrote:
An equilateral triangle is inscribed in a circle. This circle is inscribed in another equilateral triangle. What is the ratio of the perimeter of the large triangle to that of the small one.


perimeter of equilateral triangle = 3R*3^1/2 (R - external circle) and perimeter = 6r*3^1/2 (r - internal circle)=> p(small) = 3R*3^1/2, p(large) = 6R*3^1/2 => p(large) = 2*p(small).
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Re: PS-33 [#permalink] New post 11 May 2004, 23:23
lastochka wrote:
Emmanuel wrote:

perimeter of equilateral triangle = 3R*3^1/2 (R - external circle) and perimeter = 6r*3^1/2 (r - internal circle)=> p(small) = 3R*3^1/2, p(large) = 6R*3^1/2 => p(large) = 2*p(small).


Can you explain that in plainer terms?


Lastochka, if you have an equilateral triangle and consider an inscribed circle, you get: p = 6*r*[3^1/2], where r is the radius of the circle, p is perimeter of the triangle. This is true because (how to explain it in simplier words? :-)) the radius of inscribed circle r = (dist^2 - (a/2)^2)^1/2, where dist - distance between any vertex of the triangle and its center, a = length of its side. dist = a/3^1/2, since there is a triangle with sides: (dist, dist, a) and greatest angle of 120. => r = a/(2*3^1/2). Since p = 3a => p = 6r*3^1/2.

if you have an equilateral triangle which is CIRCUMscribed in a circle, then if R - its radius, a is the length of the side of this trianle, => R = a/3^1/2. The logis is similar: R = dist(which was defined previously), R = a/(3^1/2). => p = 3*a {sum of all sides} = 3*R*3^1/2.

The solution is straightforward now, since we have 1 circle, which is INscrideb in big triangle and at the same time a triangle which is CIRCUMscribed in that same circle. => p(big) = 6*r*3^1/2, p(small) = 3*r*3^1/2.

I know, this solution lacks illustratin, but maybe this can help you:
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Re: PS-33 [#permalink] New post 13 May 2004, 17:17
hallelujah1234 wrote:
An equilateral triangle is inscribed in a circle. This circle is inscribed in another equilateral triangle. What is the ratio of the perimeter of the large triangle to that of the small one.


I am not sure if I got the Q right. It says there are 2 equilateral triangles one inside a circle and one outside the came circle .

For the one inside the perimeter comes out to be 3*(2RCos30)
For the one outside the perimeter is 3*(2R/Tan30)

Hence Ratio= Sin 30 = 0.5 ??
Re: PS-33   [#permalink] 13 May 2004, 17:17
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