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In the above diagram, the circle inscribes the larger equilateral, [#permalink]
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08 Jun 2018, 06:22
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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?
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In the above diagram, the circle inscribes the larger equilateral, [#permalink]
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08 Jun 2018, 07:24
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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 306090 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)
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In the above diagram, the circle inscribes the larger equilateral, [#permalink]
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08 Jun 2018, 07:40
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.



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In the above diagram, the circle inscribes the larger equilateral, [#permalink]
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08 Jun 2018, 08:35
GMATPrepNow wrote: 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? Excellent approaches have already posted by pushpitkcTo 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.<<<< PropertyWe 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)
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Re: In the above diagram, the circle inscribes the larger equilateral, [#permalink]
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08 Jun 2018, 09:12
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 circleTherefore, 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
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Re: In the above diagram, the circle inscribes the larger equilateral, [#permalink]
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10 Jun 2018, 07:34
GMATPrepNow wrote: 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 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 = 2So, each side of the smaller equilateral triangle has length 2Using this information, we can create a 306090 triangle (in blue) We can now compare this blue 306090 triangle with the BASE 306090 triangleBy the property of similar triangles, we know that the ratios of corresponding sides will be equal. That is: 1/ √3 = r/ 2Cross 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 At this point, we can focus our attention on the GREEN 306090 triangle 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 reuse 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
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In the above diagram, the circle inscribes the larger equilateral, [#permalink]
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10 Jun 2018, 07:52
GMATPrepNow wrote: 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 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 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. 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 rangeNow 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
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Re: In the above diagram, the circle inscribes the larger equilateral, [#permalink]
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10 Jun 2018, 23:19
GMATPrepNow wrote: 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? Area of smaller triangle = \sqrt{3} = 1\frac{2[}{fraction] x b x h Area of larger triangle = 1[fraction]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[}{fraction] x 4 x b x h = 4 x 1[fraction]2} x b x h = 4 \sqrt{3} Answer is BKudos if it helped..!




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