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# If ∠B, ∠ADB are 90o, A=60o, and C=30o, what is the area of the triangl

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Math Revolution GMAT Instructor
Joined: 16 Aug 2015
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If ∠B, ∠ADB are 90o, A=60o, and C=30o, what is the area of the triangl  [#permalink]

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20 Jul 2017, 01:04
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25% (medium)

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82% (01:54) correct 18% (02:46) wrong based on 34 sessions

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If ∠B, ∠ADB are $$90^o$$, $$A=60^o$$, and $$C=30^o$$, what is the area of the triangle BDC?

1) The perimeter of the triangle ABD is 3+√3.
2) The perimeter of the triangle BCD is 3+3√3.

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"Only $149 for 3 month Online Course" "Free Resources-30 day online access & Diagnostic Test" "Unlimited Access to over 120 free video lessons - try it yourself" Senior CR Moderator Status: Long way to go! Joined: 10 Oct 2016 Posts: 1354 Location: Viet Nam Re: If ∠B, ∠ADB are 90o, A=60o, and C=30o, what is the area of the triangl [#permalink] ### Show Tags 20 Jul 2017, 01:53 MathRevolution wrote: Attachment: The attachment 7.20.png is no longer available If ∠B, ∠ADB are $$90^o$$, $$A=60^o$$, and $$C=30^o$$, what is the area of the triangle BDC? 1) The perimeter of the triangle ABD is 3+√3. 2) The perimeter of the triangle BCD is 3+3√3. Attachment: 7.20.png [ 4.01 KiB | Viewed 594 times ] Assume that $$AD = x$$, so we have $$AB = 2AD = 2x$$ Also, we have $$AC = 2 AB = 4x \implies CD = 3x$$ In triangle ABD, we have $$BD = \sqrt{AB^2 - AD^2} = \sqrt{4x^2 - x^2} = x\sqrt{3}$$ In triangle ABC, we have $$BC = 2BD = 2x\sqrt{3}$$ The area of the triangle BCD is $$\frac{1}{2} \times x\sqrt{3} \times 3x = \frac{3\sqrt{3} x^2}{2}$$. Hence we simply need to know the value of $$x$$ to know the area of the triangle BCD. (1) The perimeter of the triangle ABD is $$3+\sqrt{3}$$ so we have $$x + 2x + x\sqrt{3} = 3+\sqrt{3} \implies x(3+\sqrt{3})=3+\sqrt{3} \implies x=1$$. Sufficient. (2) The perimeter of the triangle BCD is $$3+3\sqrt{3}$$ so we have $$x\sqrt{3} + 2x\sqrt{3} + 3x = 3+3\sqrt{3} \implies x(3\sqrt{3} + 3) = 3+3\sqrt{3} \implies x=1$$. Sufficient. The answer is D. _________________ Math Revolution GMAT Instructor Joined: 16 Aug 2015 Posts: 7226 GMAT 1: 760 Q51 V42 GPA: 3.82 Re: If ∠B, ∠ADB are 90o, A=60o, and C=30o, what is the area of the triangl [#permalink] ### Show Tags 23 Jul 2017, 18:24 ==> In the original condition, if AD=x, you get $$BD=\sqrt{3}x, AB=2x$$, and from $$BC=2\sqrt{3}x$$ and AC=4x, there is 1 variable (x). In order to match the number of variables to the number of equations, there must be 1 equation. Since there is 1 for con 1) and 1 for con 2), D is most likely to be the answer. By solving con 1) and con 2) separately, you get con 1)=con 2), which is x=1, hence it is unique and sufficient. The answer is D. Answer: D _________________ MathRevolution: Finish GMAT Quant Section with 10 minutes to spare The one-and-only World’s First Variable Approach for DS and IVY Approach for PS with ease, speed and accuracy. "Only$149 for 3 month Online Course"
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Re: If ∠B, ∠ADB are 90o, A=60o, and C=30o, what is the area of the triangl   [#permalink] 23 Jul 2017, 18:24
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