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If the area of ΔABC is , what is the length of AB?

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If the area of ΔABC is , what is the length of AB? [#permalink]

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New post 27 Dec 2015, 10:39
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If the area of ΔABC is \(8\sqrt{3}\), what is the length of AB?

A. 4
B. 5
C. 6
D. 7
E. 8

[Reveal] Spoiler:
Attachment:
2015-12-27_2136.png
2015-12-27_2136.png [ 8.96 KiB | Viewed 1373 times ]
[Reveal] Spoiler: OA

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Re: If the area of ΔABC is , what is the length of AB? [#permalink]

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New post 27 Dec 2015, 22:21
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Bunuel wrote:
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If the area of ΔABC is , what is the length of AB?

A. 4
B. 5
C. 6
D. 7
E. 8

[Reveal] Spoiler:
Attachment:
2015-12-27_2136.png


I believe the area is missing from the question stem

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Re: If the area of ΔABC is , what is the length of AB? [#permalink]

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New post 30 Dec 2015, 11:41
Hi Bunuel,

The prompt is missing the total area (and while there are plenty of smart people here, correctly answering this question without that information will be difficult).

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Re: If the area of ΔABC is , what is the length of AB? [#permalink]

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Re: If the area of ΔABC is , what is the length of AB? [#permalink]

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New post 30 Dec 2015, 23:04
\(1/2 * BC * AB = 8\sqrt{3} => BC * AB = 16\sqrt{3}\)

\(BC = \frac{Hyp}{2} and AB = \frac{Hyp}{2 *} \sqrt{3}\)

Comparing
\(\frac{x}{2} * \frac{x}{2}\sqrt{3} = 4 * 4\sqrt{3} =>\frac{x}{2} * \frac{x}{2} \sqrt{3} = \frac{8}{2} * \frac{8}{2}\sqrt{3}\)

Therefore x=8

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If the area of ΔABC is , what is the length of AB? [#permalink]

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New post 31 Dec 2015, 00:53
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Expert's post
Bunuel wrote:
Image
If the area of ΔABC is \(8\sqrt{3}\), what is the length of AB?

A. 4
B. 5
C. 6
D. 7
E. 8

[Reveal] Spoiler:
Attachment:
2015-12-27_2136.png


Area = \(\frac{1}{2}\)*AC*BC

tan 60 = AC/BC
\(\sqrt{3}\) BC = AC

Area = \(\frac{1}{2}\)\(\sqrt{3}\) BC * BC = \(8\sqrt{3}\)
Hence BC = 4

Cos 60 = BC/AB
AB = BC/Cos 60 = BC/(1/2) = 2BC = 8

Option E
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Re: If the area of ΔABC is , what is the length of AB? [#permalink]

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New post 02 Jan 2016, 07:43
This is an 30- 60 -90 triangle so the lengths of the sides will be i nratio:
x(sqrt(3)) : x: 2x

Area given is 8(sqrt(3))
1/2 * x(sqrt(3)) * x = 8(sqrt(3))

thus x = 4
since BC = 2x => 8

Ans:E

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Re: If the area of ΔABC is , what is the length of AB? [#permalink]

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New post 12 Aug 2017, 14:43
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If the area of ΔABC is , what is the length of AB? [#permalink]

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New post 12 Aug 2017, 15:25
Bunuel wrote:
Image
If the area of ΔABC is \(8\sqrt{3}\), what is the length of AB?

A. 4
B. 5
C. 6
D. 7
E. 8

[Reveal] Spoiler:
Attachment:
2015-12-27_2136.png

The missing angle A must be 30°

I have a hard time tracking on side names, so I use terms such as "short leg."

30-60-90 right triangles have sides in ratio

short leg: long leg: hypotenuse

\(x: x\sqrt{3}: 2x\)

Let short leg BC = base = x

Let long leg AC = height=\(x\sqrt{3}\)

Area Δ = \(\frac{b*h}{2}\), given as \(8\sqrt{3}\)

\(\frac{x * x\sqrt{3}}{2}\)=\(8\sqrt{3}\)

\(16\sqrt{3}\) = \(x * x\sqrt{3}\)

Divide by \(\sqrt{3}\)

16 = x\(^2\)
x = 4

That's the short leg. Hypotenuse AB length is twice that, 2x = 8.

Answer E

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Re: If the area of ΔABC is , what is the length of AB? [#permalink]

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New post 12 Aug 2017, 20:49
Area ABC = 1/2*AC*BC
so 8√3 * 2 = AB * BC = 16√3
the relation of triangle sides is 1 to √3 to 2
then x * x√3 = (x^2)√3 = 16√3
so x = BC = 4
then AC = 4√3

=>> AC^2+BC^2=AB^2
(4√3)^2+4^2=64
AB = √64 = 8

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Re: If the area of ΔABC is , what is the length of AB?   [#permalink] 12 Aug 2017, 20:49
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