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

Question

https://i.imgur.com/sd4EcIB.png

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

jayantbakshi · Answer

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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PKN · Answer

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)

pushpitkc · Answer

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)

BrentGMATPrepNow · Answer

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....
 https://i.imgur.com/4s2wG5R.png

Now drop a line down like this to create a SPECIAL 30-60-90 right triangle
 https://i.imgur.com/aLVDzG4.png

Since the base 30-60-90 right triangle has lengths 1, 2 and √3, let's give the triangle these same measurements...
 https://i.imgur.com/9R9BADP.png

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...
 https://i.imgur.com/SwexqSp.png 
.... 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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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

Prep Toolkit

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