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circle circumscribed about

Problem Solving — algebra and arithmetic (GMAT Focus Edition)
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circle circumscribed about

by sidon » Sat Apr 24, 2010 1:26 am
Hi,
I need help with the following :

A circle circumscribed about an equilateral triangle of side 'a' . the diamter of the circle is ?

my answer was a/(3^1/2) , the test answer was 2a/(3^1/2) .

Can some one help - let me see where I am wrong ?

thanks,
Sidon
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by rockeyb » Sat Apr 24, 2010 3:00 am
sidon wrote:Hi,
I need help with the following :

A circle circumscribed about an equilateral triangle of side 'a' . the diamter of the circle is ?

my answer was a/(3^1/2) , the test answer was 2a/(3^1/2) .

Can some one help - let me see where I am wrong ?

thanks,
Sidon


The triangle is an equilateral triangle with sides "a" .

The triangle is circumscribed by a circle .

The height or the altitude of the equilateral triangle with side "a" is (sqrt3/ 2)a .

Now since the triangle is an equilateral triangle the circumcenter that is the center of the circle around the triangle and the orthocenter that is the point of intersection of 3 altitudes of the triangle are same .

Also the orthocenter divides the height of the triangle in the ratio 1:3 .

So radius of the circle = 2/3 of the height .

So radius = 2/3 [(sqrt3/ 2)a] = sqrt3 x a / 3 .

So diameter = 2a x sqrt3/ 3 this can be written as 2a / sqrt3 .

Hope this helps .
"Know thyself" and "Nothing in excess"
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Junior | Next Rank: 30 Posts
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by sidon » Sat Apr 24, 2010 5:24 am
thanks , you are right .
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