GMAT Boost Challenge Question #3

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gmatboost: Master | Next Rank: 500 Posts; Posts: 312; Joined: Tue Aug 02, 2011 3:16 pmFollowed by: 33 members; Location: New York City; Thanked: 130 times; GMAT Score:780

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

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Source: Beat The GMAT — Problem Solving | Fri Aug 26, 2011 11:34 am

force5: Legendary Member; Posts: 582; Joined: Tue Mar 08, 2011 12:48 amFollowed by: 6 members; Thanked: 61 times; GMAT Score:740

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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rppala90: Senior | Next Rank: 100 Posts; Posts: 34; Joined: Mon Nov 15, 2010 12:12 pm; Thanked: 2 times

by rppala90 » Fri Aug 26, 2011 12:18 pm

3 . sqrt(3) / 4 . pi

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gmatboost: Master | Next Rank: 500 Posts; Posts: 312; Joined: Tue Aug 02, 2011 3:16 pmFollowed by: 33 members; Location: New York City; Thanked: 130 times; GMAT Score:780

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

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robosc9: Junior | Next Rank: 30 Posts; Posts: 26; Joined: Thu Dec 23, 2010 3:38 pm

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...

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varungoel: Junior | Next Rank: 30 Posts; Posts: 10; Joined: Mon Sep 12, 2011 7:24 pm

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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