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Geometry - Triangle

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Geometry - Triangle

by singhmaharaj » Tue May 06, 2014 5:54 am
In the figure [/img] see attachment, if the area of the tiangle on the right is twice the area of the triangle on the left, then in terms of s, S =

A) Sqrt 2 (s) / 2

B) Sqrt 3 (s) /2

C) Sqrt 2 (s)

D) Sqrt 3 (s)

E) 2s[/list]
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by [email protected] » Tue May 06, 2014 10:34 am
Hi singhmaharaj,

This question can be beaten by TESTing Values. Here's how....

Since we're not given any side lengths or angles to work with, we can TEST easy values and track the results.

I'm going to make the small triangle a 3/4/5 right triangle.

Area = (B)(H)/2 = (3)(4)/2 = 12/2 = 6

We're told that the big triangle has TWICE the area of the small triangle, so...

Area = 12

These triangles are SIMILAR triangles though, so the relative side lengths are all the same proportion bigger.

So, the small triangle is 3/4/5

To have twice the area, the big triangle must be...

3(root2)/4(root2)/5(root2)

This area = (3root2)(4root2)/2 = (3)(4)(2)/2 = 12

Thus, side S in the bigger triangle is equal to side s in the smaller triangle multiplied by (root2)

Final Answer: C

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by GMATGuruNY » Tue May 06, 2014 10:39 am
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Since each triangle has the same combination of angles, the triangles are SIMILAR.
The combination of angles is not specified, so we can plug in our own values.
Let each triangle be a 45-45-90 triangle so that the sides are proportioned x : x : x√2.

Smaller triangle:
Let s=1.
Thus, b = h = 1.
Area = 1/2bh = 1/2*1*1 = 1/2.

Larger triangle:
Since the larger triangle is twice as big, area = 2 * 1/2 = 1.
In the larger triangle, B = H = S.
Since the area is equal to 1, we get:
(1/2)S² = 1
S = √2. This is our target.

Now plug s=1 into the answers to see which yields our target of √2.

Only answer choice C works:
√2s = √2*1 = √2.

The correct answer is C.
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by Brent@GMATPrepNow » Tue May 06, 2014 10:43 am
singhmaharaj wrote:In the figure [/img] see attachment, if the area of the tiangle on the right is twice the area of the triangle on the left, then in terms of s, S =

A) (√2)s/2

B) (√3)s/2

C) (√2)s

D) (√3)s

E) 2s
When two triangles are similar, the length of EACH SIDE of the larger triangle is some factor (multiplier) greater than its corresponding side of the smaller triangle.
So, if we say that the length of each side of the larger triangle is k TIMES the length of each corresponding side of the triangle, we can write S = sk
Likewise, if we let H = the height of the large triangle and let h = the height of the small triangle, then we can write H = hk

We're told the following: (area of large triangle) = 2(area of small triangle)
Area = (1/2)(base)(height), so we can write: (1/2)SH = 2(1/2)sh
Divide both sides by 1/2 to get: SH = 2sh
Now substitute to get: (sk)(hk) = 2sh
Simplify: shk² = 2sh
Divide both sides by sh to get: k² = 2
Solve, to get k = √2

Great! So, the length of each side of the larger triangle is √2 the length of each corresponding side of the triangle.

So, the length of side S = (√2)s

Answer: C

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by gmat_chanakya » Tue May 06, 2014 1:58 pm
This is simple right:

For similar triangles, if the sides are in ratio a:b, then the areas are in ratio a^2:b^2.

In this case lets assume a is the side of the small triangle and b is the side of the larger triangle.
We know these are similar triangles and it is given that area of larger triangle is twice the are of the smaller triangle. This implies b^2 = 2 a^2 and therefore b = sqrt(2)a.

This means that any side in the bigger triangle sqrt(2) times the side of the smaller triangle. Therefore S = sqrt(2)s.

Could the above folks please validate if my approach is right ?
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