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If ABC is an equilateral triangle, and BC=4

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If ABC is an equilateral triangle, and BC=4 [#permalink] New post 03 Sep 2019, 06:34
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If ABCABC is an equilateral triangle,.png
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If \(ABC\) is an equilateral triangle, and \(BC=4\sqrt{3}\), what is the approximate length
of one side of square \(WXYZ\)?

A) 1.9
B) 2.9
C) 3.2
D) 4.1
E) 4.6
[Reveal] Spoiler: OA

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Last edited by Carcass on 27 Nov 2019, 11:28, edited 2 times in total.
Edited by Carcass
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Re: If ABC is an equilateral triangle, and BC=4 [#permalink] New post 03 Sep 2019, 06:37
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GreenlightTestPrep wrote:
Attachment:
Z04.png


If \(ABC\) is an equilateral triangle, and \(BC=4\sqrt{3}\), what is the approximate length
of one side of square \(WXYZ\)?

A) 1.9
B) 2.9
C) 3.2
D) 4.1
E) 4.6


Since \(ABC\) is an equilateral triangle, we know the following angles are 60° each.
Also, let's let n = the length of each side of the square
Image


Since BWX is also an equilateral triangle, we know that all 3 sides have length n:
Image


Since \(BC=4\sqrt{3}\), and since \(BX = n\), we know that side \(XC=4\sqrt{3}-n\)
Image


At this point, we can see that triangle XYC is a special 30-60-90 right triangle.
Image
When we compare ∆XYC with the base 30-60-90 triangle, we can compare corresponding sides to create the following equation: (4√3 - n)/2 = n/√3
Cross multiply to get: (√3)(4√3 - n)= (2)(n)
Simplify to get: 12 - (√3)n = 2n
Add (√3)n to both sides to get: 12 = 2n + (√3)n
Factor right side to get: 12 = n(2 + √3)
Divide both sides by (2 + √3) to get: n = 12/(2 + √3)

PRO TIP #1: By test day, all students should have the following approximations memorized:
√2 ≈ 1.4
√3 ≈ 1.7
√5 ≈ 2.2
So, 12/(2 + √3) ≈ 12/(2 + 1.7) ≈ 12/3.7

PRO TIP #2: We need not calculate the actual value of 12/3.7
Instead, notice that 12/3 = 4 and 12/4 = 3
Since 3.7 is BETWEEN 3 and 4, we now that 12/3.7 must be between 3 and 4
In other words, 12/3.7 = 3.something.

Answer: C

Cheers,
Brent
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Re: If ABC is an equilateral triangle, and BC=4 [#permalink] New post 02 Dec 2019, 14:17
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Could anyone give feedback on my approach?
So my logic was that the corner X of the square was fairly in the middle of segment BC.
Then,
4(sqrt(3)) is approximately 6.9
6.9/2 is 3.45

The closest option we have is 3.2 (option C).
Re: If ABC is an equilateral triangle, and BC=4   [#permalink] 02 Dec 2019, 14:17
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If ABC is an equilateral triangle, and BC=4

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