union can be a star

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sanju09: GMAT Instructor; Posts: 3650; Joined: Wed Jan 21, 2009 4:27 amFollowed by: 80 members; Location: India; Thanked: 267 times; GMAT Score:760

by sanju09 » Mon Mar 07, 2011 1:19 am

When two congruent equilateral triangles share a common center, then their union can be a star as shown. If their overlap is a regular hexagon of area 60, then what is the area of one of the original equilateral triangles?
(A) 10
(B) 15
(C) 30
(D) 60
(E) 90

The mind is everything. What you think you become. -Lord Buddha

Sanjeev K Saxena
Quantitative Instructor
The Princeton Review - Manya Abroad
Lucknow-226001

www.manyagroup.com

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Source: Beat The GMAT — Problem Solving | Mon Mar 07, 2011 1:19 am

Haldiram Bhujiawala: Junior | Next Rank: 30 Posts; Posts: 29; Joined: Thu Mar 11, 2010 3:23 am; Thanked: 1 times

by Haldiram Bhujiawala » Mon Mar 07, 2011 1:54 am

IMO15

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Anurag@Gurome: GMAT Instructor; Posts: 3835; Joined: Fri Apr 02, 2010 10:00 pmFollowed by: 523 members; Location: Milpitas, CA; Thanked: 1854 times; GMAT Score:770

by Anurag@Gurome » Mon Mar 07, 2011 2:23 am

sanju09 wrote:When two congruent equilateral triangles share a common center, then their union can be a star as shown. If their overlap is a regular hexagon of area 60, then what is the area of one of the original equilateral triangles?
(A) 10
(B) 15
(C) 30
(D) 60
(E) 90

Easiest Solution:
Area of any of the original triangle must be greater than the overlapping region. Only such option is 90, i.e. option E.

Methodical Tricky Solution:
Join the center of the triangles (also of the hexagon) with the vertices of the hexagon. This will result 6 small equilateral triangles inside the hexagon, each of equal are as shown in the following figure.

Now, each of these six small equilateral triangles are congruent to each of the six equilateral triangles outside the hexagon.

Now, area of hexagon = 6*(Area of an small equilateral triangle) = 60
=> Area of an small equilateral triangle = 60/6 = 10

Area of one of the original equilateral triangle = 9*(Area of an equilateral small triangle) = 90

Correct answer is E.

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sanju09: GMAT Instructor; Posts: 3650; Joined: Wed Jan 21, 2009 4:27 amFollowed by: 80 members; Location: India; Thanked: 267 times; GMAT Score:760

by sanju09 » Mon Mar 07, 2011 3:32 am

Anurag@Gurome wrote:
sanju09 wrote:When two congruent equilateral triangles share a common center, then their union can be a star as shown. If their overlap is a regular hexagon of area 60, then what is the area of one of the original equilateral triangles?
(A) 10
(B) 15
(C) 30
(D) 60
(E) 90
Easiest Solution:
Area of any of the original triangle must be greater than the overlapping region. Only such option is 90, i.e. option E.

Methodical Tricky Solution:
Join the center of the triangles (also of the hexagon) with the vertices of the hexagon. This will result 6 small equilateral triangles inside the hexagon, each of equal are as shown in the following figure.

Now, each of these six small equilateral triangles are congruent to each of the six equilateral triangles outside the hexagon.

Now, area of hexagon = 6*(Area of an small equilateral triangle) = 60
=> Area of an small equilateral triangle = 60/6 = 10

Area of one of the original equilateral triangle = 9*(Area of an equilateral small triangle) = 90

Correct answer is E.

your easiest solution is much appreciated because it also serves the purpose of the answer choices for why they are so made.

The mind is everything. What you think you become. -Lord Buddha

Sanjeev K Saxena
Quantitative Instructor
The Princeton Review - Manya Abroad
Lucknow-226001

www.manyagroup.com

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Haldiram Bhujiawala: Junior | Next Rank: 30 Posts; Posts: 29; Joined: Thu Mar 11, 2010 3:23 am; Thanked: 1 times

by Haldiram Bhujiawala » Mon Mar 07, 2011 11:47 am

Unfortunately I got the diagram wrong.

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