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In the equilateral triangle below, each side has a length of 4 units.
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05 Dec 2014, 07:41
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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:
20141205_1839.png [ 4.62 KiB  Viewed 3207 times ]
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Re: In the equilateral triangle below, each side has a length of 4 units.
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05 Dec 2014, 10:45
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



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Re: In the equilateral triangle below, each side has a length of 4 units.
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05 Dec 2014, 12:54
In the equilateral triangle PRS, each side has a length of 4 units.
Therefore, area of the equilateral triangle PRS = \(4^2*\sqrt{3}/4 = 4 * \sqrt{3}\)
In the right angle triangle \(\triangle PQT\), PQ has a length of 1 unit and TQ is perpendicular to PR.
Also, in the right angle triangle \(\triangle\) PQT, the angle QPT and angle PTQ are \(60^{\circ}\) and \(30^{\circ}\) respectively.
Therefore using the property of 306090 triangle, \(PT = 2\) and \(QT = \sqrt{3}\)
So, area of the triangle \(\triangle\) PQT= \(\sqrt{3}\)/2
Area of the region QRST = Area of equilateral triangle PRS  Area of the right angle triangle PQT
=\(\sqrt{3}*{4} \sqrt{3}/2\) \(= \sqrt{3}* 7/2\)
Answer: C



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Re: In the equilateral triangle below, each side has a length of 4 units.
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08 Dec 2014, 06:10
Bunuel wrote: 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? Attachment: 20141205_1839.png 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 The correct answer is C.
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Re: In the equilateral triangle below, each side has a length of 4 units.
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08 Dec 2014, 19:10
Answer = C) \(\frac{7}{2} \sqrt{3}\) Attachment:
20141205_1839.png [ 6.19 KiB  Viewed 2973 times ]
Area of equilateral triangle \(= \frac{\sqrt{3}}{4} 4^2 = 4\sqrt{3}\) Area of Right Angle triangle (306090) (Sides \(1  \sqrt{3}  2\)) \(= \frac{1}{2} * 1 * \sqrt{3} = \frac{1}{2} \sqrt{3}\) Area of shaded region \(= 4\sqrt{3}  \frac{1}{2} \sqrt{3} = \frac{7}{2} \sqrt{3}\)
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Re: In the equilateral triangle below, each side has a length of 4 units.
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31 Mar 2019, 02:12
∆QTP ; 30:60:90 side PT= 2 and QT = √3; area of ∆QTP = √3/2 area of ∆PRS; 4√3 so area of side QRTS = 4√3√3/2 IMO C \(\frac{7}{2} \sqrt{3}\) Bunuel wrote: 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: 20141205_1839.png



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Re: In the equilateral triangle below, each side has a length of 4 units.
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10 Jun 2019, 05:45
Area of the quadrilateral=area of the triangle PRS area of the triangle PQT By applying the the rule of 1:sqrt3:2 in the 30:60:90 triangle PQT, we get the height QT=sqrt3. now the area of PRS area of PQT 4 * sqrt3  sqrt3/2 (7* sqrt3)/2 Choice C
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Re: In the equilateral triangle below, each side has a length of 4 units.
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