Summer is Coming! Join the Game of Timers Competition to Win Epic Prizes. Registration is Open. Game starts Mon July 1st.

It is currently 20 Jul 2019, 11:23

Close

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
Your Progress

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

Not interested in getting valuable practice questions and articles delivered to your email? No problem, unsubscribe here.

Close

Request Expert Reply

Confirm Cancel

If each side of ΔACD above has length 3 and if AB has length

  new topic post reply Question banks Downloads My Bookmarks Reviews Important topics  
Author Message
TAGS:

Hide Tags

Find Similar Topics 
Manager
Manager
avatar
Status: May The Force Be With Me (D-DAY 15 May 2012)
Joined: 06 Jan 2012
Posts: 197
Location: India
Concentration: General Management, Entrepreneurship
Reviews Badge
If each side of ΔACD above has length 3 and if AB has length  [#permalink]

Show Tags

New post Updated on: 28 Dec 2018, 06:44
3
20
00:00
A
B
C
D
E

Difficulty:

  45% (medium)

Question Stats:

73% (02:33) correct 27% (02:50) wrong based on 308 sessions

HideShow timer Statistics


Attachment:
Equilateral.png
Equilateral.png [ 1.79 KiB | Viewed 8381 times ]
If each side of ΔACD above has length 3 and if AB has length 1, what is the area of region BCDE?

(A) \(\frac{9}{4}\)

(B) \(\frac{7}{4} \sqrt{3}\)

(C) \(\frac{9}{4} \sqrt{3}\)

(D) \(\frac{7}{2} \sqrt{3}\)

(E) \(6 + \sqrt{3}\)

Project PS Butler : Question #108


Subscribe to get Daily Email - Click Here | Subscribe via RSS - RSS

_________________
Giving +1 kudos is a better way of saying 'Thank You'.

Originally posted by boomtangboy on 07 Apr 2012, 20:39.
Last edited by HKD1710 on 28 Dec 2018, 06:44, edited 2 times in total.
Edited the question and the diagram
Most Helpful Expert Reply
Math Expert
User avatar
V
Joined: 02 Sep 2009
Posts: 56304
Re: If each side of ΔACD above has length 3 and if AB has length  [#permalink]

Show Tags

New post 08 Apr 2012, 02:34
6
3
boomtangboy wrote:
Attachment:
The attachment Equilateral.png is no longer available
If each side of ΔACD above has length 3 and if AB has length 1, what is the area of region BCDE?

(A) \(\frac{9}{4}\)

(B) \(\frac{7}{4} \sqrt{3}\)

(C) \(\frac{9}{4} \sqrt{3}\)

(D) \(\frac{7}{2} \sqrt{3}\)

(E) \(6 + \sqrt{3}\)

Attachment:
Equilateral.png
Equilateral.png [ 1.79 KiB | Viewed 8706 times ]

Since each side of ΔACD has length 3 then ACD is an equilateral triangle and its each angle is 60°.

Now, the are of equilateral triangle is \(area_{equilateral}=side^2*\frac{\sqrt{3}}{4}=9\frac{\sqrt{3}}{4}\) (for more check math-triangles-87197.html);

Next, since angle A is 60° then right triangle ABE is a 30°-60°-90° right triangle. Now, in a right triangle where the angles are 30°, 60°, and 90° the sides are always in the ratio \(1 : \sqrt{3}: 2\), the leg opposite 30° (AB) corresponds with \(1\) and the leg opposite 60° (BE) corresponds with \(\sqrt{3}\) (the hypotenuse AE corresponds with 2). So, since \(AB=1\) then \(BE=\sqrt{3}\), and the area of ABE is \(\frac{1}{2}*AB*BE=\frac{\sqrt{3}}{2}\);

The area of of region BCDE is the area of ACD minus the area of ABE: \(9\frac{\sqrt{3}}{4}-\frac{\sqrt{3}}{2}=9\frac{\sqrt{3}}{4}-2\frac{\sqrt{3}}{4}=7\frac{\sqrt{3}}{4}\).

Answer: B.

Hope it's clear.
_________________
General Discussion
Director
Director
avatar
Status: There is always something new !!
Affiliations: PMI,QAI Global,eXampleCG
Joined: 08 May 2009
Posts: 950
Re: If each side of ΔACD above has length 3 and if AB has length  [#permalink]

Show Tags

New post 07 Apr 2012, 23:08
2
equilateral triangle hence Angle A = 60 deg
thus, in Triangle ABE, AB = 1 BE = 3^(1/2) and AE = 2

Area BCDE = Area ACD - Area ABE

Area ACD = 3^(1/2)/4 * (side)^2

Area ABE = 1/2 * 3^(1/2) * 1

we get 3.5/2 * 3^(1/2) multiplying numerator and denominator by 2 we get B
Manager
Manager
User avatar
G
Joined: 22 May 2015
Posts: 123
Re: If each side of ΔACD above has length 3 and if AB has length  [#permalink]

Show Tags

New post 31 Dec 2018, 20:51
In Triangle ABE one angle is 90, angle BAE is 60 (as ACD is equilateral) hence other angle is 30.

Sin 30 = 1/2 , Sin 60 = Root 3 / 2

Sin 30 = side opposite to the 30 degrees / Hypotenuse
Sin 60 = side opposite to the 60 degree / Hypotenuse

Solving we get Ab = 1 , AE = 2 , BE = 1

Area of ABE = 1/2 * 1 * Root 3

BCDE = ACD - ABE
solving we get option B
_________________
Consistency is the Key
e-GMAT Representative
User avatar
V
Joined: 04 Jan 2015
Posts: 2943
Re: If each side of ΔACD above has length 3 and if AB has length  [#permalink]

Show Tags

New post 02 Jan 2019, 04:09

Solution


Given:
    • In triangle ACD, each side length = 3
    • AB = 1

To find:
    • The area of quadrilateral BCDE

Approach and Working:
Area of BCDE = area of ACD – area of ABE
    • Area of ACD = \(\frac{√3 * 3^2}{4} = \frac{9√3}{4}\)
    • Area of ABE = \(\frac{1}{2} * 1 * BE = \frac{BE}{2}\)

And, we know angle A = 60 degrees. So, we can say that ABE is a 30 – 60 – 90 degrees triangle.
    • Thus, AB : BE : AE = 1 : √3 : 2
      o Implies, BE = √3

    • So, area of ABE = \(\frac{√3}{2}\)

Therefore, area of BCDE = \(\frac{9√3}{4} - \frac{√3}{2} = \frac{7√3}{4}\)

Hence, the correct answer is Option B

Answer: B

Image

_________________
VP
VP
User avatar
D
Joined: 09 Mar 2016
Posts: 1273
Re: If each side of ΔACD above has length 3 and if AB has length  [#permalink]

Show Tags

New post 19 Jan 2019, 13:11
what is the formula of area of an irregular quadrilateral ?
Senior PS Moderator
User avatar
D
Status: It always seems impossible until it's done.
Joined: 16 Sep 2016
Posts: 751
GMAT 1: 740 Q50 V40
GMAT 2: 770 Q51 V42
GMAT ToolKit User Reviews Badge
Re: If each side of ΔACD above has length 3 and if AB has length  [#permalink]

Show Tags

New post 19 Jan 2019, 13:30
1
Hi dave13, How have you been? :-)

No - there is no such formula ( at least not that I know of, or within the scope of GMAT)

So, this question is testing three things -
1. Area of an equilateral triangle. (Also the fact that all angles are equal to 60 degrees in one)
2. 30-60-90 triangle side ratios
3. Area of a right-angled triangle.

My thought process -
Okay. We need to find the area of an unual looking quadrilateral. Let's try to find the area of the two regular triangles that we know of and subtract the smaller one from the larger triangle to find the unusual area.
The side of the equilateral triangle is 3 so its area is \(\frac{s^2}{4}*\sqrt{3}\) remember this - comes handy :-)
\(\frac{3^2}{4}*\sqrt{3}\)
\(\frac{9}{4}*\sqrt{3}\)


The smaller triangle is 30-60-90 (one angle is 90 degrees and one angle is common between itself and the equilateral triangle)

Okay, so the side ratios of a 30-60-90 triangle are 1, \(\sqrt{3}\) and 2 with the smallest side being opposite the smallest angle.
This implies area of right angle triangle is -
\(\frac{1}{2}b*h\)
\(\frac{1}{2}*\sqrt{3}\)

So the area we are interested in will be the difference -
\(\frac{9}{4}*\sqrt{3} - \frac{1}{2}*\sqrt{3}\)
\(\frac{7}{4}*\sqrt{3}\)


Hope this helps. Have a nice day. :-)
dave13 wrote:
what is the formula of area of an irregular quadrilateral ?

_________________
Regards,
Gladi



“Do. Or do not. There is no try.” - Yoda (The Empire Strikes Back)
VP
VP
avatar
G
Joined: 09 Mar 2018
Posts: 1002
Location: India
Re: If each side of ΔACD above has length 3 and if AB has length  [#permalink]

Show Tags

New post 16 Feb 2019, 11:59
boomtangboy wrote:
Attachment:
Equilateral.png
If each side of ΔACD above has length 3 and if AB has length 1, what is the area of region BCDE?

(A) \(\frac{9}{4}\)

(B) \(\frac{7}{4} \sqrt{3}\)

(C) \(\frac{9}{4} \sqrt{3}\)

(D) \(\frac{7}{2} \sqrt{3}\)

(E) \(6 + \sqrt{3}\)

Project PS Butler : Question #108


Subscribe to get Daily Email - Click Here | Subscribe via RSS - RSS


Now since its an equilateral triangle area = \(\frac{9}{4} \sqrt{3}\)

area of smaller triangle = 1/2 * 1 * \(\sqrt{3}\), since it will be a 30-60-90 triangle

Area of region will be

\(\frac{9}{4} \sqrt{3}\) - 1/2 * 1 * \(\sqrt{3}\)

\(\frac{7}{4} \sqrt{3}\)
_________________
If you notice any discrepancy in my reasoning, please let me know. Lets improve together.

Quote which i can relate to.
Many of life's failures happen with people who do not realize how close they were to success when they gave up.
GMAT Club Bot
Re: If each side of ΔACD above has length 3 and if AB has length   [#permalink] 16 Feb 2019, 11:59
Display posts from previous: Sort by

If each side of ΔACD above has length 3 and if AB has length

  new topic post reply Question banks Downloads My Bookmarks Reviews Important topics  





Powered by phpBB © phpBB Group | Emoji artwork provided by EmojiOne