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05 Dec 2014, 07:41
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C
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Tough and Tricky questions: Geometry.
In the equilateral triangle below, each side has a length of 4 units. If PQ has a length of 1 unit and TQ is perpendicular to PR, what is the area of region QRST?
A) \(\frac{1}{3} \sqrt{3}\)
B) \(3 \sqrt{3}\)
C) \(\frac{7}{2} \sqrt{3}\)
D) \(4 \sqrt{3}\)
E) \(15 \sqrt{3}\)
Kudos for a correct solution.
Source: Chili Hot GMAT
Attachment:
2014-12-05_1839.png [ 4.62 KiB | Viewed 6759 times ]
08 Dec 2014, 19:10
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Answer = C) \(\frac{7}{2} \sqrt{3}\)
2014-12-05_1839.png [ 6.19 KiB | Viewed 6329 times ]
Area of equilateral triangle \(= \frac{\sqrt{3}}{4} 4^2 = 4\sqrt{3}\)
Area of Right Angle triangle (30-60-90) (Sides \(1 - \sqrt{3} - 2\)) \(= \frac{1}{2} * 1 * \sqrt{3} = \frac{1}{2} \sqrt{3}\)
Area of shaded region \(= 4\sqrt{3} - \frac{1}{2} \sqrt{3} = \frac{7}{2} \sqrt{3}\)
Attachment:
2014-12-05_1839.png [ 6.19 KiB | Viewed 6329 times ]
Area of equilateral triangle \(= \frac{\sqrt{3}}{4} 4^2 = 4\sqrt{3}\)
Area of Right Angle triangle (30-60-90) (Sides \(1 - \sqrt{3} - 2\)) \(= \frac{1}{2} * 1 * \sqrt{3} = \frac{1}{2} \sqrt{3}\)
Area of shaded region \(= 4\sqrt{3} - \frac{1}{2} \sqrt{3} = \frac{7}{2} \sqrt{3}\)
General Discussion
intheend14
GMAT 1: 740 Q49 V41
Posts: 125
05 Dec 2014, 10:45
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Sorry for not having pictures, but hopefully the explanation will suffice.
The area of the big triangle (Triangle PRS) can be calculated by the area of an equilateral triangle, A = side^2*sqrt(3)/4. This yields 4*sqrt(3) for the big triangle.
Now for the smaller triangle, let's name this Triangle PQT with angles p, q and t (just to abbreviate). We know angle p = 60 degrees (it's one of the angles in the large equilateral triangle), angle q = 90 degrees as QT is perpendicular to PR, which means that angle t = 30 degrees. From this information, we have a 30:60:90 triangle with sides in the ratio of 1:sqrt(3):2. From this, the area of Triangle PQT = 1/2*PQ*QT = 1/2*1*sqrt(3) = sqrt(3)/2.
Now if we subtract the area of Triangle PQT from the area of Triangle PRS, we get the area of the shape QRST, which is 4sqrt(3) - 0.5sqrt(3) = 3.5*sqrt(3).
Choice C
The area of the big triangle (Triangle PRS) can be calculated by the area of an equilateral triangle, A = side^2*sqrt(3)/4. This yields 4*sqrt(3) for the big triangle.
Now for the smaller triangle, let's name this Triangle PQT with angles p, q and t (just to abbreviate). We know angle p = 60 degrees (it's one of the angles in the large equilateral triangle), angle q = 90 degrees as QT is perpendicular to PR, which means that angle t = 30 degrees. From this information, we have a 30:60:90 triangle with sides in the ratio of 1:sqrt(3):2. From this, the area of Triangle PQT = 1/2*PQ*QT = 1/2*1*sqrt(3) = sqrt(3)/2.
Now if we subtract the area of Triangle PQT from the area of Triangle PRS, we get the area of the shape QRST, which is 4sqrt(3) - 0.5sqrt(3) = 3.5*sqrt(3).
Choice C
