gmatt1476 wrote:
In the figure above, the shaded region represents the front of an upright wooden frame around the entrance to an amusement park ride. If \(RS = \frac{5\sqrt{3}}{2}\) meters, what is the area of the front of the frame?
(1) x = 9 meters
(2) \(ST = 2\sqrt{3}\) meters
DS07402.01
In many shaded area problems, the shaded area can be calculated indirectly as the difference between some big and small areas. Here, both the big and the small areas are given by equilateral triangles.
We know that \(RS=5\sqrt{3}/2\). The original question: A(shaded)=A(big)-A(small)=?
1) We know that x=9. The formula for the height of an equilateral triangle with side length a is \(h=\frac{\sqrt{3}}{2}a\). Thus, knowing either the side length or the height of an equilateral triangle is enough to calculate its area with the usual formula for triangles, area=(base)(height)/2.
x is the side length of the big equilateral triangle, so we could determine A(big). The heigth of the small equilateral triangle is the difference between the height of the big equilateral triangle and RS, and we could determine this difference, which would lead us to A(small). Thus, we could get a unique value to answer the original question. \(\implies\)
Sufficient2) We know that \(ST=2\sqrt{3}\), which is the height of the small equilateral triangle. The height of the big equilateral triangle is RS+ST, both of which are known. Thus, we could get a unique value to answer the original question. \(\implies\)
SufficientAnswer: D
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