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If ABCD is a square, and XYZ is an equilateral triangle,
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08 Jul 2018, 05:18
08 Jul 2018, 05:18
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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
*I'll post a solution in 2 days
Re: If ABCD is a square, and XYZ is an equilateral triangle,
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08 Jul 2018, 19:45
08 Jul 2018, 19:45
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GreenlightTestPrep wrote:
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
*I'll post a solution in 2 days
Here,
Let the side of the equilateral △ = a
and radius of the circle be = r
we know,
area of equilateral △ = \(\frac{\sqrt3}{4} * side^2\)
or area of equilateral △ = \(\frac{\sqrt3}{4} * a^2\)
If equilateral △ is inscribed inside a circle then the side of the equilateral △ =\(\sqrt3 * radius of the circle\)
i.e. \(a = \sqrt3 * r\)(substituting the values)
or \(r = \frac{a}{\sqrt3}\)
Now the side of the square = diameter of the circle = \(2 * radius = 2r = 2 * \frac{a}{\sqrt3}\)
Area of the square = \((\frac{2a}{\sqrt3})^2\)
Therefore,
\(\frac{Area of the square}{Area of the triangle}= (\frac{2a}{\sqrt3})^2 * \frac{4}{(\sqrt3 * a^2)} = \frac{16}{(3\sqrt3)} = \frac{(16\sqrt3)}{9}\)
i.e. Area of the square = \(\frac{(16\sqrt3)}{9}\) times the area of the triangle
Retired Moderator
Joined: 10 Apr 2015
Posts: 6218
Given Kudos: 136
Re: If ABCD is a square, and XYZ is an equilateral triangle,
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10 Jul 2018, 04:41
10 Jul 2018, 04:41
GreenlightTestPrep wrote:
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
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....
Now drop a line down like this to create a SPECIAL 30-60-90 right triangle
Since the base 30-60-90 right triangle has lengths 1, 2 and √3, let's give the triangle these same measurements...
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...
.... 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
