In terms of x and y, what is the area of the shaded region?

Question

https://gmatclub.com/forum/download/file.php?id=25997 In terms of x and y, what is the area of the shaded region? Note: Figure not drawn to scale A. x^2 - y^2 B. \sqrt{3}(x^2 - y^2) C. \sqrt{3}x^2 - y^2 D. \frac{(x^2 -y^2)}{2} E. \frac{\sqrt{3}(x^2 - y^2)}{2} Kudos for a correct solution . nestedtr_q1.png

Bunuel · Accepted Answer

MAGOOSH OFFICIAL SOLUTION: nestedtr_explanation.png

bhatiamanu05 · Answer

Consider triangle which is having base Y as A
Consider triangle which is having base X as B

Using similar triangle rule
so hypotenuse of A = 2y.
so perpendicular = sqrt(4y^2 - Y^2) = sqrt(3)y
so area of triangle A = 1/2*y*sqrt(3)y = sqrt(3)/2*y^2

similarly Area of triangle B = sqrt(3)/2*x^2

so area of shaded = sqrt(3)/2(x^2 - y^2)

ngie · Answer

The big triangle is a 30:60:90 and so is the small, white one.
Given  A_{x} = Area of the big triangle and  A_{y} =area of the small triangle
we have that 
 A_{x}=x\cdot\sqrt{3}x=x^2\cdot\sqrt{3} 
and  A_{y}=y\cdot\sqrt{3}y=y^2\cdot\sqrt{3}

The shaded area equals the difference:  \sqrt{3}(x^2-y^2)

B is the correct answer

EMPOWERgmatRichC · Answer

Hi ngie,

You have the right conceptual idea, but you have to be very careful with your work.

The formula for area of a triangle is (1/2)(Base)(Height).

Your calculations don't use the correct formula.

GMAT assassins aren't born, they're made,
Rich

camlan1990 · Answer

The area of un-shaded region = y^2/x^2 * The area of the big triangle => The area of shaded region = 1 - y^2/x^2

The area of the big triangle = 1/2*x^2*sqrt(3) => The area of shaded one is 1/2*sqrt(3)*(x^2 - y^2)

=> ANSWER: E

peachfuzz · Answer

Both triangles are 30 60 90 triangles since they are similar triangles.
I just subtracted the larger triangle from the smaller one.
(1/2)(x^2)(x*sqrt(3)) - (1/2)(y)(y*sqrt(3))
 \frac{\sqrt{3}(x^2 - y^2)}{2}

Answer: E

ngie · Answer

You are right! Thank you for pointing it out!

PareshGmat · Answer

Shaded region is a trapezoid nestedtr_explanation.png There is a direct formula to calculate the area once we have the height of trapezoid = \sqrt{3}(x-y) Area = \frac{1}{2} (x+y) \sqrt{3}(x-y) = \frac{\sqrt{3} (x^2 - y^2)}{2} Answer = E

iliavko · Answer

How are we sure that we can use the 30 60 90 proportions? We only know that its a right triangle. Plus 2x, x ratio. does it mean that we must have a 30 60 90?

Thanks!

Bunuel · Answer

MUST KNOW FOR THE GMAT:
 • A right triangle where the angles are 30°, 60°, and 90°. 
 https://gmatclub.com/forum/download/file.php?id=10157 
This is one of the 'standard' triangles you should be able recognize on sight. A fact you should commit to memory is:  The sides are always in the ratio  1 : \sqrt{3}: 2 . 
Notice that the smallest side (1) is opposite the smallest angle (30°), and the longest side (2) is opposite the largest angle (90°).

• A right triangle where the angles are 45°, 45°, and 90°.  
 https://gmatclub.com/forum/download/file.php?id=10158 
This is one of the 'standard' triangles you should be able recognize on sight. A fact you should also commit to memory is: The sides are always in the ratio  1 : 1 : \sqrt{2} . With the  \sqrt{2}  being the hypotenuse (longest side). This can be derived from Pythagoras' Theorem. Because the base angles are the same (both 45°) the two legs are equal and so the triangle is also isosceles.

adkikani · Answer

Bunuel   niks18   gmatbusters   amanvermagmat   pushpitkc

Is my below understanding correct?

For the larger triangle, I know base x and hypotenuse (2x) hence I could find height using ratio 30:60:90 (angles)
vs sides (1: \sqrt{3} : 2)
But in smaller triangle (i.e with base y) only three angles and base y is known, Hence we derived height as  \sqrt{3y}

amanvermagmat · Answer

Hello If your understanding is that smaller triangle is also similar to the larger triangle and thus the three angles of this smaller right angle triangle will be 30-60-90 and thus height of this smaller triangle will be √3*y,.. .. then it's correct

Posted from my mobile device

GMATBusters · Answer

your question : Is my below understanding correct? For the larger triangle, I know base x and hypotenuse (2x) hence I could find height using ratio 30:60:90 (angles) No, . In fact the the sides x and 2x let you know that the triangle is 30:60:90 triangle. All 30-60-90-degree triangles have sides with the same basic ratio. So only a single side is required to find the value of all sides. A 30-60-90-degree right triangle. 369580.image1.jpg

adityasuresh · Answer

In the is question the ratio of lengths of sides are mention in the stem as x:2x and from the figure it’s mentioned that it’s a right angle so we think of a 30-60-90 triangle whose sides are in the ratio x:root3x:2x.

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In terms of x and y, what is the area of the shaded region?

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In terms of x and y, what is the area of the shaded region?

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