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Math Expert V
Joined: 02 Sep 2009
Posts: 58347
In the equilateral triangle below, each side has a length of 4 units.  [#permalink]

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Difficulty:   25% (medium)

Question Stats: 78% (02:38) correct 22% (03:04) wrong based on 133 sessions

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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 3207 times ]

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Re: In the equilateral triangle below, each side has a length of 4 units.  [#permalink]

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1
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.  [#permalink]

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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 30-60-90 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

=$$\sqrt{3}*{4}- \sqrt{3}/2$$ $$= \sqrt{3}* 7/2$$

Math Expert V
Joined: 02 Sep 2009
Posts: 58347
Re: In the equilateral triangle below, each side has a length of 4 units.  [#permalink]

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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:
2014-12-05_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.  [#permalink]

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Answer = C) $$\frac{7}{2} \sqrt{3}$$

Attachment: 2014-12-05_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 (30-60-90) (Sides $$1 - \sqrt{3} - 2$$) $$= \frac{1}{2} * 1 * \sqrt{3} = \frac{1}{2} \sqrt{3}$$

Area of shaded region $$= 4\sqrt{3} - \frac{1}{2} \sqrt{3} = \frac{7}{2} \sqrt{3}$$
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Re: In the equilateral triangle below, each side has a length of 4 units.  [#permalink]

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∆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:
2014-12-05_1839.png
Manager  B
Joined: 24 Dec 2011
Posts: 52
Location: India
GPA: 4
WE: General Management (Health Care)
Re: In the equilateral triangle below, each side has a length of 4 units.  [#permalink]

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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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