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If ABCD is a square, and XYZ is an equilateral triangle, then the area

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If ABCD is a square, and XYZ is an equilateral triangle, then the area  [#permalink]

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New post 08 Jul 2018, 05:10
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Question Stats:

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If ABCD is a square, and XYZ is an equilateral triangle, then the area of the square is how many times the area of the triangle?

A) (4√3)/3
B) (8√3)/3
C) 2√6
D) (16√3)/9
E) (16√2)/3

*kudos for all correct solutions

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Re: If ABCD is a square, and XYZ is an equilateral triangle, then the area  [#permalink]

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New post 08 Jul 2018, 06:09
D.. let radius of circle be x.
Area of square: 4x^2
Maximum side of an equilateral triangle in a circle of radius x is: x root 3.
Calculate area of triangle. Easy straightforward solution.


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If ABCD is a square, and XYZ is an equilateral triangle, then the area  [#permalink]

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New post 08 Jul 2018, 06:27
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GMATPrepNow wrote:
Image

If ABCD is a square, and XYZ is an equilateral triangle, then the area of the square is how many times the area of the triangle?

A) (4√3)/3
B) (8√3)/3
C) 2√6
D) (16√3)/9
E) (16√2)/3

*kudos for all correct solutions


Let the side of equilateral triangle be x unit.
So, area of equilateral triangle =\(\sqrt{3}*x^2/4\)
We know that side of inscribed equilateral triangle is \(\sqrt{3}\) times the radius of circle.
so, x=\(\sqrt{3}*r\) where 'r' is the radius of circle.
Now we have side of square=diameter of circle=2r=2*\(\frac{x}{\sqrt{3}}\)
So, area of square=\((\frac{2x}{\sqrt{3}})^2\)

Now \(\frac{A_{square}}{A_{Triangle}}\)=\((\frac{2x}{\sqrt{3}})^2\)/(\(\sqrt{3}/4\)*\(x^2\))=\(\frac{16}{3\sqrt{3}}\)= \((16\sqrt{3})/9\)
Ans. (D)
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If ABCD is a square, and XYZ is an equilateral triangle, then the area  [#permalink]

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New post 08 Jul 2018, 10:52
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GMATPrepNow wrote:
Image

If ABCD is a square, and XYZ is an equilateral triangle, then the area of the square is how many times the area of the triangle?

A) (4√3)/3
B) (8√3)/3
C) 2√6
D) (16√3)/9
E) (16√2)/3

*kudos for all correct solutions


Formula used: Area of equilateral triangle = \(\frac{√3}{4}*a^2\) where a - side of the triangle

If the equilateral triangle has an area of 4√3, the side of the equilateral triangle is \(\frac{a^2}{4} = 4\) -> \(a = 4\)

The relation between the side of the equilateral triangle and the radius of the circle circumscribing it is \(a = r*√3\)

Now r = \(\frac{4}{√3}\) and the side of the square is twice the radius of the circle. Area of square is \((\frac{2*4}{√3})^2 = \frac{64}{3}\)

Therefore the area of the square is \(\frac{\frac{64}{3}}{4√3} = \frac{64}{12√3} = \frac{16}{3√3} = \frac{(16√3)}{9}\) times the area of the equilateral triangle (Option D)
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Re: If ABCD is a square, and XYZ is an equilateral triangle, then the area  [#permalink]

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New post 10 Jul 2018, 04:39
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GMATPrepNow wrote:
Image

If ABCD is a square, and XYZ is an equilateral triangle, then the area of the square is how many times the area of the triangle?

A) (4√3)/3
B) (8√3)/3
C) 2√6
D) (16√3)/9
E) (16√2)/3

*kudos for all correct solutions


If ∆XYZ is EQUILATERAL, then each angle is 60°
So, if we draw a line from the center to a vertex, we'll get two 30° angles....
Image

Now drop a line down like this to create a SPECIAL 30-60-90 right triangle
Image

Since the base 30-60-90 right triangle has lengths 1, 2 and √3, let's give the triangle these same measurements...
Image

IMPORTANT: This means the circle's radius = 2, which means the circle's DIAMETER = 4
Notice that the circle's diameter = the length of one side of the square
So, each side of the square has length 4, which means the area of the square = (4)(4) = 16

Okay, now let's determine the area of the triangle

Since we know that one side of the special 30-60-90 right triangle has length √3...
Image
.... we know that the length of one side of the equilateral triangle = 2√3

This allows us to apply a special area formula for equilateral triangles:
Area of equilateral triangle = (√3)(side²)/4

So, the area of ∆XYZ = (√3)(2√3)²/4
= (√3)(12)/4
= 3√3

The area of the square is how many times the area of the triangle?
Answer = 16/3√3
Check the answer choices...not there!

Multiply top and bottom by √3 to get: (16√3)/9
Answer: D

Cheers,
Brent
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Re: If ABCD is a square, and XYZ is an equilateral triangle, then the area &nbs [#permalink] 10 Jul 2018, 04:39
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