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Re: In the figures above, if the area of the triangle on the [#permalink]

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12 Mar 2013, 00:44

1

This post received KUDOS

fozzzy wrote:

Please, can anyone help out in this particular problem? Thanks!

Bunuel has already provided a solution. Nonetheless, assume the given triangles to be equilateral. The area of the first triangle is\(\sqrt{3}/4*s^2\). The area of the other triangle would be \(\sqrt{3}/4*S^2\). Given that\(\sqrt{3}/4*S^2\) = \(2*\sqrt{3}/4*s^2\). Thus, S = \(\sqrt{2}s.\)
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Re: In the figures above, if the area of the triangle on the [#permalink]

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15 Mar 2013, 14:26

You must know that property for GMAT:

"Similar triangles and areas: the ratio of the areas of two similar triangles is the square of the ratio of corresponding lengths" and the inverse... "The ratio of the lengths of two similar triangles is the square root of the ratio of corresponding areas"

(Area left)x2=(Area right) meaning that the ratio of the areas is \(\frac{Area_{right}}{Area_{left}}=2\)

Therefore, the ratio of the lengths: \(\frac{S}{s}=\sqrt{2}\)

The concept of similar triangles is relatively rare on the GMAT (you likely won't see it more than once on Test Day and you probably won't see it at all). That having been said, the concept is about the ratios of the sides (and how side length effects other things - area, perimeter, other sides, etc.).

Here, we have two triangles with the exact same set of 3 angles, so we know that the two triangles are similar. The one ratio that we're given to work with is that the area of the larger triangle is exactly TWICE that of the smaller triangle. We can use this ratio, along with TESTing VALUES, to get to the solution.

Rather than deal with an abstract triangle, I'm going to say that the small triangle is a 3/4/5 right triangle...

Small Triangle Area = (1/2)(Base)(Height) Area = (1/2)(3)(4) = 6

Since the larger triangle has TWICE this area, we know that it's area is 6(2) = 12...

Large Triangle Area = (1/2)(Base)(Height) 12 = (1/2)(Base)(Height)

Since the triangles are similar, each side of the larger triangle is the same proportionate larger than the corresponding side of the smaller triangle. This means that the two sides (the 3 and the 4) have to each be multiplied by the same number and the result has to DOUBLE the area. The only way to get to DOUBLE is if each side is multiplied by \sqrt{2}

12 = (1/2)(3\sqrt{2})(4\sqrt{2})

In this way, when you multiply everything, you get....

I turned the triangles into right isosceles triangles, the smaller one with legs "1" and 1", which gives an area of 1/2. This would mean the larger triangle would have an area of 1 (double a half). This would make each leg of the larger triangle \(\sqrt{2}\)?

I turned the triangles into right isosceles triangles, the smaller one with legs "1" and 1", which gives an area of 1/2. This would mean the larger triangle would have an area of 1 (double a half). This would make each leg of the larger triangle \(\sqrt{2}\)?

Much appreciated!

Since a PS question can have only one correct answer then considering one particular case which satisfies the conditions given must also give correct answer.

So, yes, if the smaller triangle is \(1:1:\sqrt{2}\), then the larger triangle is \(\sqrt{2}:\sqrt{2}:2\), thus \(s=\sqrt{2}\) and \(S=s*\sqrt{2}=\sqrt{2}*\sqrt{2}=2\).
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