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705-805 (Hard)|   Geometry|      
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BrentGMATPrepNow
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If ABC is an equilateral triangle, and BC=4*3^(1/2), what is the appro 02 Sep 2019, 08:04
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C
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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


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C
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If ABC is an equilateral triangle, and BC=4*3^(1/2), what is the appro 02 Sep 2019, 10:12
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I am bad at this, but I came up with C. We know the length of a side of the big triangle is 4root3 or approx 6.8. The length of one side of the square plus AW has to equal approx 6.8. you can use a ratio to determine the proportion of how much of that 6.8 should be the square and AW. For example, using the 30/60/90 formulae if x were 2 you would have 4 + 2root3 to add up to what would the equivalent of a large triangle side in this problem. The ratio I came up with was a little less than 1/2(about 7/15) for what the side of the square should be. so a little less than 6.8 is 3.2.
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If ABC is an equilateral triangle, and BC=4*3^(1/2), what is the appro 02 Sep 2019, 15:15
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Assume BX=WX=XY=x
and CX =y

XY= sin(60) * CX
\(XY\)= \(\sqrt{3}y/2\)

\(y=\frac{2x}{\sqrt{3}}\)

BC=x+y=\(4\sqrt{3}\)

\(x+\frac{2x}{\sqrt{3}}\)=\(4\sqrt{3}\)
\(x=\frac{12}{2+\sqrt{3}}\)
x≈3.2




GMATPrepNow


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


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hb42f3J.png
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If ABC is an equilateral triangle, and BC=4*3^(1/2), what is the appro 03 Sep 2019, 07:27
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GMATPrepNow


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


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hb42f3J.png
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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



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



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



At this point, we can see that triangle XYC is a special 30-60-90 right triangle.

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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If ABC is an equilateral triangle, and BC=4*3^(1/2), what is the appro 04 Sep 2019, 11:39
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Height of Equilateral Triangle is √3/2 * (length of the side); the height of Triangle ABC = 6.
Let, the side of the square WXYZ be "n", then the side of the triangle BWX will also be "n".
Height of Triangle BWX is √3/2 * (n).
Height of ABC = Height of BWX + Height of Square
6=√3/2 * (n)+n
or, n= 12/(√3+2)
or, n = 12/3.7≈3.2

Answer: C
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If ABC is an equilateral triangle, and BC=4*3^(1/2), what is the appro 06 Sep 2019, 09:45
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GMATPrepNow


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


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Hi GMATPrepNow,
Can we not assume x to be the midpoint of side BC?
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If ABC is an equilateral triangle, and BC=4*3^(1/2), what is the appro 06 Sep 2019, 10:59
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stne
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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


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hb42f3J.png


Hi GMATPrepNow,
Can we not assume x to be the midpoint of side BC?
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Good idea, but x is not the midpoint of side BC.
Now that we know n ≈ 3.2, we can test that theory by plugging that n-value into our diagram.
When we do so, we see that BX ≠ XC

Cheers,
Brent
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If ABC is an equilateral triangle, and BC=4*3^(1/2), what is the appro 06 Sep 2019, 16:53
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GMATPrepNow


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


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Attachment:
hb42f3J.png
Show more

We can start out by setting one side of the square as \(x\) and attempt to fill out the rest of the lengths in terms of \(x\). Notice the top triangle is also equilateral so \(BX = WX = x\). Using the right triangle XYC and \(XY = x\), we get \(XC = 2 \sqrt{3}/3 * x\) by the ratio \(\frac{XY}{XC} = \sqrt{3} / 2\)

Therefore we have \(BC = BX + XC = x + 2\sqrt{3}/3* x = 4\sqrt{3}\). We may substitute in \(\sqrt{3} = 1.7\) to help us, \(BC = 6.8 = x + 1.13x = 2.13x\)
So x will be a little bigger then 3, we choose C.
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If ABC is an equilateral triangle, and BC=4*3^(1/2), what is the appro 30 Jul 2025, 04:29
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