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

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New post 20 Jul 2017, 01:04
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A
B
C
D
E

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

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New post 20 Jul 2017, 01:53
MathRevolution wrote:
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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.


Attachment:
7.20.png
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.
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Re: If ∠B, ∠ADB are 90o, A=60o, and C=30o, what is the area of the triangl  [#permalink]

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

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