What is the area of the shaded region in the figure above, in terms of

Question

https://gmatclub.com/forum/download/file.php?id=46877 What is the area of the shaded region in the figure above, in terms of a, b, and c? A. \sqrt{3}(a^2 + b^2 + c^2) B. \frac{\sqrt{3}}{2}(a^2 - b^2 - c^2) C. \frac{\sqrt{3}}{2}(a^2 - b^2 + c^2) D. \frac{\sqrt{3}}{2}(a^2 + b^2 - c^2) E. \frac{\sqrt{3}}{2}(a^2 + b^2 + c^2) trid19.png

Shef08 · Answer

How does one solve it? Bunuel, please come to the rescue

Posted from my mobile device

Archit3110 · Answer

solved using 30:60:90 rule
we get sides of the height and bases for triangles 
upon finding the sides of the triangle we can say the area of the shaded region 
  \frac{\sqrt{3}}{2}(a^2 - b^2 + c^2)

IMO C

akrockheartz19 · Answer

We are asked to find two shaded regions: Trapezium and right angle Triangle.

Using 30-60-90 rule, we can find the height of the bigger triangle, base for the smaller triangle, and the height of the trapezium.

Add the area of the two regions and the answer is C

KarishmaB · Answer

Note that the figure has 3 similar triangles with the same common angle and 90 degree angles. 
The largest triangle has hypotenuse 2a and one leg as a*sqrt(3).
So the other leg will be

\sqrt{(2a)^2 - (a \sqrt3)^2} = \sqrt{4a^2 - 3a^2} = a

So 3 corresponding sides of the 3 similar triangles are a, b and c.
Hence their areas will be in the ratio a^2 : b^2 : c^2

Now note that the shaded region will be obtained by subtracting the middle triangle from the largest triangle and adding the smallest triangle back.

So the shaded area will look like k*(a^2 - b^2 + c^2)

There is only one such option - (C)

Answer (C)

medic19 · Answer

VeritasKarishma  Karishma, hi - do not think it is necessary to set out the formula - the 1:SqR3:2 triangle rule should be pretty much ingrained in our DNA. Obviously you wouldn't use it if you took the test but just for the purposes of GMAT technique? Unless there is something else you wanted to check.

Surely closer the 700 scale? I guess looking at the answer choices we are given a clue that all will have SqR3/2 in common.

Contrast to my comment above, wondering if there is a more detailed algebraic solution?

KarishmaB · Answer

I am not sure I follow your question here. Yes, the more we ingrain the basic concepts, the easier it will get during the test. You see a 30-60-90 triangle, the ratio of sides should come to mind and vice versa. You see a right triangle with 3-4 (or multiples) as legs, 5 (or multiple) should come to mind and so on...

Sides of the largest triangle are: a, sqrt(3)*a, 2a

Area of the largest triangle = (1/2)*a * sqrt(3)*a = sqrt(3)/2 * a^2

Ratio of the areas of the three triangles = a^2 : b^2 : c^2

So area of the three triangles are sqrt(3)/2 * a^2, sqrt(3)/2 * b^2 and sqrt(3)/2 * c^2

So the shaded region = (sqrt(3)/2) (a^2 - b^2 + c^2)

Answer (C)

bumpbot · Answer

Automated notice from GMAT Club BumpBot: 

A member just gave Kudos to this thread, showing it’s still useful. I’ve bumped it to the top so more people can benefit. Feel free to add your own questions or solutions.

 This post was generated automatically.

What is the area of the shaded region in the figure above, in terms of

Prep Toolkit

Top 5 GMAT Debrief Videos

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards

GMAT Problem Solving (PS) Questions

What is the area of the shaded region in the figure above, in terms of

Prep Toolkit

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards