Author Topic: Problems  (Read 1205 times)

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Offline Ender

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Problems
« on: February 27, 2008, 06:21:19 am »
I will post problems here when I feel like it.

Offline Ender

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Re: Problems
« Reply #1 on: February 27, 2008, 06:24:39 am »
Problem 1. Source: aops

A square and equilateral triangle are circumscribed around a common circle. What is the ratio of their areas?

Offline rabbit

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Re: Problems
« Reply #2 on: February 27, 2008, 08:09:32 am »
Jew!  Wait...

Offline d&q

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Re: Problems
« Reply #3 on: February 27, 2008, 09:13:00 pm »
Since the triangle is equilateral, the center of the circle is not only the incenter but the centroid. Therefore, the height of the triangle is in a 3:1 ratio with the radius of the circle. Assuming a unit circle, the height is 3, and the area is 3*sqrt(3). The square's area is 1.

Triangle:Square = 3sqrt(3)/4.

[Edit]: Haha fuck, the squares area is 4, lulz.
« Last Edit: February 27, 2008, 09:15:57 pm by Deuce »
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Offline Ender

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Re: Problems
« Reply #4 on: February 28, 2008, 03:41:30 am »
Since the triangle is equilateral, the center of the circle is not only the incenter but the centroid. Therefore, the height of the triangle is in a 3:1 ratio with the radius of the circle. Assuming a unit circle, the height is 3, and the area is 3*sqrt(3). The square's area is 1.

Triangle:Square = 3sqrt(3)/4.

[Edit]: Haha fuck, the squares area is 4, lulz.

Correct!