ratios of triangle and square

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clinton: Junior | Next Rank: 30 Posts; Posts: 18; Joined: Tue Feb 02, 2010 9:26 am; Thanked: 1 times

ratios of triangle and square

by clinton » Tue May 31, 2011 7:10 pm

An equilateral triangle with each side T and a square with the perimeters s have the same area. What is the ratio of T:S

1. 2:3
2. 16:3
3. 4:square root of 3
4. 2: 4(square root of 3)
5. 4: 4(square root of 3)

Sorry there's no visual but, I'm wondering if someone could help with the question. Where ended up was:

area of triangle = are of square

1/2(b)(h) = s^2

=1/2(t)(3t^/2) = s^2 I used pythagorean theorm to find the height?

Please Help!!!

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Source: Beat The GMAT — Problem Solving | Tue May 31, 2011 7:10 pm

SoCan: Senior | Next Rank: 100 Posts; Posts: 97; Joined: Sun May 15, 2011 9:19 am; Thanked: 18 times; Followed by:1 members

by SoCan » Tue May 31, 2011 7:37 pm

A good formula to know is the area of an equilateral triangle.

The heigh of an equilateral triangle is (t/2)(sqr(3)). If you draw a line from a vertex to the midpoint of the opposite side, you create two 30-60-90 triangles, which have set ratios for the length of a side. For such a triangle, given shortest side t, the second longest side is t(sqr(3)) and the longest side is 2t. This second longest side is the height. However, since we are given that t is the full length of a triangle, the ratios must apply to t/2, not t. Thus, the height of an equilateral triangle is (t/2)(sqr(3)).

Now we have the base and the height, so we can plug in for b*h/2, which gives the area of an equilateral triangle:
(t^2)(sqr(3))/4 --> this is easy to memorize and you'll save time if you run across a question that deals with the area of such a triangle

The length of a square with perimeter s is s/4. Therefore the area is (s/4)^2 = (s^2)/16

Set the two equations equal to each other
(t^2)(sqr(3))/4=(s^2)/16
(t^2)/(s^2)=4/(16sqr(3)

take the root of both sides

t/s = 2/(4(fourthroot(3))
t/s = 1/(2(fourthroot(3))

So I don't get an answer that matches. It's closest to answer 4, but it looks like the question writer took the square of the 16 but not the sqr(3).

I don't think I did this wrong, but would like to hear from others. What's the source of the question?

Last edited by SoCan on Tue May 31, 2011 8:12 pm, edited 2 times in total.

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by Anurag@Gurome » Tue May 31, 2011 7:52 pm

clinton wrote:An equilateral triangle with each side T and a square with the perimeters s have the same area. What is the ratio of T:S

1. 2:3
2. 16:3
3. 4:square root of 3
4. 2: 4(square root of 3)
5. 4: 4(square root of 3)

Sorry there's no visual but, I'm wondering if someone could help with the question. Where ended up was:

area of triangle = are of square

1/2(b)(h) = s^2

=1/2(t)(3t^/2) = s^2 I used pythagorean theorm to find the height?

Please Help!!!

Solution:
The area of the equilateral triangle is {sqrt(3)/4} * (T^2).
Area of square is (S/4)^2.
Now, {sqrt(3)/4} * (T^2) = (S/4)^2.
So, T^2 : S^2 = 1/{sqrt(3)*4}.
Or, T:S = 1/{2*(3^[1/4])}.

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arjuncr: Newbie | Next Rank: 10 Posts; Posts: 2; Joined: Mon Sep 20, 2010 11:50 pm

by arjuncr » Tue May 31, 2011 11:03 pm

base of triangle - T
height of triangle equilateral = sqrt (3)/2 * base = {sqrt(3)/2}*T
The area of the equilateral triangle - 1/2 (base)* (height) = 1/2 *(T) *({sqrt(3)/2}*T).
Now,
Perimeter of square = S .
Therefore one side will be - S/4 . (square has 4 equal sides.)

Therefore Area of square is (S/4)^2.
Now, 1/2 *(T) *({sqrt(3)/2}*T) = (S/4)^2.
simplifying ,
sqrt(3)/4 *T^2 = 1/16 *S^2
Or, T^2:S^2 = (1/4) *(sqrt(3)}. => T:S = 1/2 * sqrt( sqrt(3))
None of the answers match ..

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