In the above diagram, the circle inscribes the larger equilateral,

Question

https://i.imgur.com/BywBeDq.png  
In the above diagram, the circle inscribes the larger equilateral, and it circumscribes the smaller equilateral triangle. If the area of the smaller triangle is √3, what is the area of the larger triangle?

A) 9π - 16√3
B) 4√3
C) 8√3
D) 16√3
E) 16π - 2√3

*kudos for all correct solutions

NOTE: There are at least 2 very different approaches we can take to solve this question. How many approaches can you find?

PKN · Accepted Answer

Excellent approaches have already posted by  pushpitkc

To add to the above, the below mentioned approach may be seen:-

Refer to the diagram affixed herewith, 
 
 All triangles are equilateral triangles and circles are inscribed in these triangles. If the sides of triangle ABC =a unit , then the side of the triangle DEF= \frac{a}{2}  unit and the side of the triangle GHI= \frac{a}{4}  unit and so on.  <<<<-------  Property

We are given, area of smaller triangle DEF= \sqrt{3}  sq. unit
 Or, (\sqrt{3}/4) * (\frac{a}{2})^2 =  \sqrt{3} 
 Or,  (\sqrt{3}/4) *  \frac{a^2}{4} =  \sqrt{3} 
 Or,  (\sqrt{3}/4) * a^2 =4* \sqrt{3} 
 Or, Area of larger triangle ABC= 4*\sqrt{3}  sq. unit

Hence option (B)

pushpitkc · Answer

Triangle_Circle.png Given: Area of smaller equilateral triangle is \sqrt{3} Formula used: Area = \frac{\sqrt{3}}{4} * a^2 | Height(h) = \frac{\sqrt{3}}{2} * a | Radius on incircle(i) = \frac{2}{3} * h where a - side of equilateral triangle Since area of the triangle is \sqrt{3} , \frac{\sqrt{3}}{4} * a^2 = \sqrt{3} -> a = 2 Height of the triangle(h) = \frac{\sqrt{3}}{2}* a = \sqrt{3} | Radius of circumcircle = \frac{2}{3} * h = \frac{2}{3} * \sqrt{3} = \frac{2}{\sqrt{3}} OR = OD = \frac{2}{\sqrt{3}} In triangle OAD 1. OAD = 30 degrees, ODA = 90 degree (We have a 30-60-90 triangle) 2. Sides of the triangle are in ratio 1:\sqrt{3}:2 Since OD = \frac{2}{\sqrt{3}} , Length of AD = 2 . AC = 2*AD = 2*2 = 4 Area of the triangle ABC = \frac{\sqrt{3}}{4} * AC^2 = \frac{\sqrt{3}}{4} * 4^2 = 4\sqrt{3} (Option B)

Blueblu · Answer

If you imagine and invert the smaller triangle inside the circle, you will note that it divides the larger triangle in 4 equal triangles. The area of the larger triangle is therefore 4 times the area of the smaller triangle. Hence B is the answer.

PP777 · Answer

Given: Area of the smaller triangle = √3

To find: Area of the larger triangle

Approach: Let side of the smaller equilateral triangle= a
Area of the smaller triangle =√3 = (√3/4) a^2 
 => a= 2

Radius of the circumcircle of an equilateral triangle= Side of the equilateral triangle/√3 = 2/√3

For a circle inscribed in an equilateral triangle, Side of the triangle = (2√3)* Radius of the circle

Therefore, Side of the larger triangle = (2√3)*(2/√3) = 4

Area of the larger triangle= (√3/4)* Side of the larger triangle = (√3/4) *4 ^2 = 4√3

Correct Answer= B

BrentGMATPrepNow · Answer

We're told that the area of the smaller triangle is √3
USEFUL FORMULA:   Area of an equilateral triangle = (√3)(side²)/4   
So, we can write: (√3)(side²)/4 = √3
Divide both sides by √3 to get: (side²)/4 = 1
Multiply both sides by 4 to get: side² = 4
Solve:  side = 2 
So, each side of the smaller equilateral triangle has length  2

Using this information, we can create a 30-60-90 triangle (in  blue )
 https://i.imgur.com/0l3yoZY.png

We can now compare this blue 30-60-90 triangle with the   BASE 30-60-90 triangle  
 https://i.imgur.com/9gbJBuY.png 
By the property of similar triangles, we know that the ratios of corresponding sides will be equal. 
That is:  1 /  √3   =  r /  2  
Cross multiply to get: (√3)(r) = 2
Solve: r = 2/√3
So, the RADIUS of the circle = 2/√3
We'll add this information to our diragram
 https://i.imgur.com/i7hnBL6.png

At this point, we can focus our attention on the GREEN 30-60-90 triangle
 https://i.imgur.com/PlAWErA.png 
Since we already know that the RADIUS of the circle = 2/√3, we can apply the property of similar triangles again. 
The ratios of corresponding sides will be equal. 
So, we get: (  2/√3  )/  1   = (  x  )/  √3   
Cross multiply to get: (2/√3)(√3) = (x)(1)
Simplify: x = 2
Since x = HALF the length of one side of the larger triangle, we know that the ENTIRE length = 4

What is the area of the larger triangle?   
We'll re-use our formula that says: area of an equilateral triangle = (√3)(side²)/4
Area = (√3)(4²)/4
= (√3)(16)/4
= 4√3

Answer: B

Cheers
Brent

BrentGMATPrepNow · Answer

Another approach is to use the following fact: 
   For GMAT Problem Solving questions (i.e., questions that are NOT data sufficiency questions), all diagrams are DRAWN TO SCALE, unless stated otherwise.     
So, we can use this to ESTIMATE the area of the larger triangle.

We know the smaller triangle has area √3
 https://i.imgur.com/w3sA94K.png 
On test day, we must memorize 3 approximations: √2 ≈ 1.4, √3 ≈ 1.7, √5 ≈ 2.2

Given this, what would you estimate the area of LARGE triangle to be?

ASIDE: If you can mentally "move" the smaller triangle to one corner, it might help your estimation. 
 https://i.imgur.com/vJWGUwV.png

If the area of the small triangle is 1.7, we might estimate the area of the large triangle to be somewhere in the   5 to 8 range

Now let's check our answer choices....

A) 9π - 16√3 ≈ (9)(3.1) - (16)(1.7) ≈  1 . This is pretty far from our estimate of   5 to 8  . ELIMINATE. 
B) 4√3 ≈ (4)(1.7) ≈  6.8 . This is within our estimated range   5 to 8  . KEEP. 
C) 8√3 ≈ (8)(1.7) ≈  13.6 . This is VERY far from our estimate of   5 to 8  . ELIMINATE. 
D) 16√3 ≈ (16)(1.7) ≈  27 . This is VERY far from our estimate of   5 to 8  . ELIMINATE. 
E) 16π - 2√3 ≈ (16)(3.1) - (2)(1.7) ≈  47 . This is VERY far from our estimate of   5 to 8  . ELIMINATE.

Answer: B

Cheers,
Brent

blueviper · Answer

Area of smaller triangle = \sqrt{3} = 1\frac{2  x b x h

Area of larger triangle = 1 2} x B x H

Now B=2b & H=2h ( Property of equilateral triangles inscribed in a circle...)

Therefore, Area of larger triangle = 1\frac{2  x 4 x b x h = 4 x 1 2} x b x h = 4 \sqrt{3}

Answer is B

Kudos if it helped..!

jfranciscocuencag · Answer

Using this information, we can create a 30-60-90 triangle (in  blue )
 https://i.imgur.com/0l3yoZY.png

We can now compare this blue 30-60-90 triangle with the   BASE 30-60-90 triangle  
 https://i.imgur.com/9gbJBuY.png 
By the property of similar triangles, we know that the ratios of corresponding sides will be equal. 
That is:  1 /  √3   =  r /  2  
Cross multiply to get: (√3)(r) = 2
Solve: r = 2/√3
So, the RADIUS of the circle = 2/√3

Hello  GMATPrepNow

I just have a question, why we can solve it like the following to find r?

x√3 = 1

So it gives me x= √3/3

Kind regards!

BrentGMATPrepNow · Answer

I believe you're calculating the enlargement factor. 
That is x = the enlargement factor

So, to find the radius, we multiply 2 (in the base triangle) by the enlargement factor to get: radius = (2)(√3/3) = 2√3/3
This is equal to the radius I calculated in my solution (2/√3)
To see why, take 2/√3 and multiply top and bottom by √3 to get 2√3/3

Does that help?

Cheers,
Brent

warrior1991 · Answer

We are given the area of smaller equilateral triangle:--√3

Finally :-
Area of bigger triangle =√3/4*A*A=(√3/4)*16=4√3

IMO B

bumpbot · Answer

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In the above diagram, the circle inscribes the larger equilateral,

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In the above diagram, the circle inscribes the larger equilateral,

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