GMATPrepNow wrote:
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
squareSince 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/
√3Cross 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/
3.7Instead, 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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