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GMAT Boost Challenge Question #3

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GMAT Boost Challenge Question #3

by gmatboost » Fri Aug 26, 2011 11:34 am
Here is a GMAT Boost Challenge Question.

On GMAT Boost, I frequently leave out the answer choices on PS questions so that the focus is on the math rather than the answers. If you have any questions, let me know.

The solution to last week's question is available here:
https://www.beatthegmat.com/gmat-boost-c ... 89063.html
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An equilateral triangle is inscribed in a circle. What fraction of the circle's area is contained within the triangle?
Greg Michnikov, Founder of GMAT Boost

GMAT Boost offers 250+ challenging GMAT Math practice questions, each with a thorough video explanation, and 100+ GMAT Math video tips, each 90 seconds or less.
It's a total of 20+ hours of expert instruction for an introductory price of just $10.
View sample questions and tips without signing up, or sign up now for full access.

Also, check out the most useful GMAT Math blog on the internet here.
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by force5 » Fri Aug 26, 2011 12:06 pm
thats area of triangle /area of circle.

Rt 3/4 side^2/Pi R^2

solving: 63/152
Approx 41.44%
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by rppala90 » Fri Aug 26, 2011 12:18 pm
3 . sqrt(3) / 4 . pi
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by gmatboost » Wed Sep 14, 2011 9:47 am
These answers are correct. The fastest way to do this without memorizing this very specific formula is to memorize these less specific formulas:

In a 30-30-120 triangle (as is created by 2 radii and one side of the inscribed equilateral triangle), the ratio of sides is x : x : x*sqrt(3).

And, the other formula you should definitely memorize is the one for area of an equilateral triangle:
https://blog.gmatboost.com/2011/08/02/me ... -triangle/

So, if the radius of the circle is r, then the side of the triangle is r*sqrt(3), and the area of the triangle is:
[r*sqrt(3)]^2 * sqrt(3)/4 = 3r^2 * sqrt(3)/4

The area of the circle is pi*r^2.

So, the fraction taken up by the triangle is
[3r^2 * sqrt(3)/4] / [pi*r^2] -> cancel out the r^2
= [3 * sqrt(3)/4] / [pi]
= [3 * sqrt(3)] / [4pi]

Check out the next challenge question:
https://www.beatthegmat.com/gmat-boost-c ... tml#406020
Greg Michnikov, Founder of GMAT Boost

GMAT Boost offers 250+ challenging GMAT Math practice questions, each with a thorough video explanation, and 100+ GMAT Math video tips, each 90 seconds or less.
It's a total of 20+ hours of expert instruction for an introductory price of just $10.
View sample questions and tips without signing up, or sign up now for full access.

Also, check out the most useful GMAT Math blog on the internet here.
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by robosc9 » Thu Sep 15, 2011 7:19 pm
gmatboost wrote:Here is a GMAT Boost Challenge Question.

An equilateral triangle is inscribed in a circle. What fraction of the circle's area is contained within the triangle?
The question is asking:



Area of the triangle
--------------------------
Area of the circle



If you can't memorize the formula (like me), it would be useful to draw a diagram for such questions.
I have attached a diagram for those who want to take a stab at it...

You will need to know...
30:60:90 rule
and
a few triangle properties
(you will have to memorize these)

Try to solve it now...

Image
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by varungoel » Sun Sep 25, 2011 4:33 pm
@ gmatboost - you mention a useful formula but the explanation isn't clear without a diagram. the manhattan gmat book doesn't talk about 30-30-120 triangle.

@robosc9 - thanks for providing the diagram.
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