traingle

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rupsk: Master | Next Rank: 500 Posts; Posts: 123; Joined: Mon Feb 07, 2011 12:11 pmFollowed by: 1 member

by rupsk » Tue Nov 15, 2011 4:58 pm

In the figure above an equilateral triangle with a perimeter of 18 inches is surrounded by a border square root of 3 inches thick. What is the area of the border?
18*square root of 3
21*square root of 3
24*square root of 3
27*square root of 3
36*square root of 3

I had solved in below way
side of small triangle =6 height=3*square root of 3
big triangle height=3*square root of 3 + 2*square root of 3=5*square root of 3
side=(2/square root of 3)*5*square root of 3

required area= (square root of 3/4)(100-36)=16*square root of 3
But there is no option available for my answer.

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Source: Beat The GMAT — Problem Solving | Tue Nov 15, 2011 4:58 pm

user123321: Master | Next Rank: 500 Posts; Posts: 385; Joined: Fri Sep 23, 2011 9:02 pmFollowed by: 6 members; Thanked: 62 times

by user123321 » Tue Nov 15, 2011 5:18 pm

for these two equilateral triangles, the centroid is same. and we know for smaller triangle
1/3rd of height of triangle gives distance from mid point of a side to centroid
=> (1/3)(root(3)/2)*6 = root(3)
for big triangle the thickness is root(3).
so for big triangle the distance from mid point to its centroid = root(3)+root(3) = 2root(3)
hence the height of big triangle = 3 * (2root(3)) = 6*root(3)
so side of big traingle = 12
big triangle area minus small triangle area = root(3)/4 * (12^2-6^2) = 27*root(3)

user123321

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Want to do it right the first time.

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rupsk: Master | Next Rank: 500 Posts; Posts: 123; Joined: Mon Feb 07, 2011 12:11 pmFollowed by: 1 member

by rupsk » Tue Nov 15, 2011 5:22 pm

please explain how you got height of big triangle.

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user123321: Master | Next Rank: 500 Posts; Posts: 385; Joined: Fri Sep 23, 2011 9:02 pmFollowed by: 6 members; Thanked: 62 times

by user123321 » Tue Nov 15, 2011 5:27 pm

rupsk wrote:please explain how you got height of big triangle.

Sure no problem.
first of all we need to know that...distance from centroid to midpoint of a side = 1/3 of height of traingle -(1)
now back to question.
for big triangle, the distance from centroid to its side = distance from centroid to small traingle's side + thickness of the strip = root(3) + root(3)

so for big traingle from (1), its height is 3*2root(3)
from this you can get side of big triangle.

And after that you can calculate difference of areas of two traingles and you can get the answer.

user123321

Just started my preparation

Want to do it right the first time.

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