GMAT Question of the Day - Daily to your Mailbox; hard ones only

 It is currently 15 Oct 2019, 21:42

### GMAT Club Daily Prep

#### Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.

Customized
for You

we will pick new questions that match your level based on your Timer History

Track

every week, we’ll send you an estimated GMAT score based on your performance

Practice
Pays

we will pick new questions that match your level based on your Timer History

# In the equilateral triangle below, each side has a length of 4 units.

Author Message
TAGS:

### Hide Tags

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

### Show Tags

05 Dec 2014, 07:41
00:00

Difficulty:

25% (medium)

Question Stats:

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

### HideShow timer Statistics

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 ]

_________________
Manager
Joined: 12 Sep 2014
Posts: 142
GMAT 1: 740 Q49 V41
GPA: 3.94
Re: In the equilateral triangle below, each side has a length of 4 units.  [#permalink]

### Show Tags

05 Dec 2014, 10:45
1
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
Manager
Joined: 04 Oct 2013
Posts: 150
Location: India
GMAT Date: 05-23-2015
GPA: 3.45
Re: In the equilateral triangle below, each side has a length of 4 units.  [#permalink]

### Show Tags

05 Dec 2014, 12:54
1
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
Joined: 02 Sep 2009
Posts: 58347
Re: In the equilateral triangle below, each side has a length of 4 units.  [#permalink]

### Show Tags

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:
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

_________________
SVP
Status: The Best Or Nothing
Joined: 27 Dec 2012
Posts: 1750
Location: India
Concentration: General Management, Technology
WE: Information Technology (Computer Software)
Re: In the equilateral triangle below, each side has a length of 4 units.  [#permalink]

### Show Tags

08 Dec 2014, 19:10
2
1
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 $$= 4\sqrt{3} - \frac{1}{2} \sqrt{3} = \frac{7}{2} \sqrt{3}$$
_________________
Kindly press "+1 Kudos" to appreciate
GMAT Club Legend
Joined: 18 Aug 2017
Posts: 4999
Location: India
Concentration: Sustainability, Marketing
GPA: 4
WE: Marketing (Energy and Utilities)
Re: In the equilateral triangle below, each side has a length of 4 units.  [#permalink]

### Show Tags

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:
2014-12-05_1839.png
Manager
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]

### Show Tags

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
_________________
------
A Reader lives a thousand lives before he dies. The one who doesn't read lives only one..

Kudos is the best way to say Thank you! Please give me a kudos if you like my post
Re: In the equilateral triangle below, each side has a length of 4 units.   [#permalink] 10 Jun 2019, 05:45
Display posts from previous: Sort by