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

It is currently 11 Nov 2019, 18:09

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

In the figure, each side of square ABCD has length 1, the length of li

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

Hide Tags

Find Similar Topics 
Manager
Manager
User avatar
P
Status: Quant Expert Q51
Joined: 02 Aug 2014
Posts: 103
Re: In the figure, each side of square ABCD has length 1, the length of li  [#permalink]

Show Tags

New post 20 Nov 2018, 09:09
hi guys,

I have prepared a detailed solution of this video.

Click here to see the explanation
_________________
Intern
Intern
avatar
B
Joined: 12 Sep 2016
Posts: 13
GMAT 1: 730 Q47 V42
WE: Accounting (Consulting)
Reviews Badge
Re: In the figure, each side of square ABCD has length 1, the length of li  [#permalink]

Show Tags

New post 17 Jan 2019, 03:44
What is the level of this question?
Math Expert
User avatar
V
Joined: 02 Sep 2009
Posts: 58954
Re: In the figure, each side of square ABCD has length 1, the length of li  [#permalink]

Show Tags

New post 17 Jan 2019, 03:49
Manager
Manager
avatar
B
Joined: 01 Jan 2019
Posts: 78
Location: Canada
Concentration: Finance, Entrepreneurship
GPA: 3.24
Re: In the figure, each side of square ABCD has length 1, the length of li  [#permalink]

Show Tags

New post 19 Feb 2019, 01:01
Bunuel wrote:
Image
In the figure, each side of square ABCD has length 1, the length of line Segment CE is 1, and the length of line segment BE is equal to the length of line segment DE. What is the area of the triangular region BCE?

A. 1/3

B. \(\frac{\sqrt{2}}{4}\)

C. 1/2

D. \(\frac{\sqrt{2}}{2}\)

E. 3/4


Image

Note that as BE=DE the triangles BCE and CDE are congruent (all three sides are equal) --> \(\angle{BCE}=\angle{DCE}\). So if we extend the line segment CE it will meet with the crosspoint of the diagonals, let's call it point O. Also note that EO will be perpendicular to BD, as diagonals in square make a right angle.

(The area of BCE) = (The area of BOE) - (The area of BOC).

Area of BOC is one fourth of the square's =\(\frac{1}{4}\).

Area BOE, \(\frac{1}{2}*BO*EO\). \(BO=\frac{\sqrt{2}}{2}\), half of the diagonal of a square. \(EO=CE+CO=1+\frac{\sqrt{2}}{2}=\frac{2+\sqrt{2}}{2}\), CO is also half of the diagonal of a square.

So \(AreaBOE=\frac{1}{2}*BO*EO=\frac{1}{2}*\frac{\sqrt{2}}{2}*\frac{2+\sqrt{2}}{2}=\frac{\sqrt{2}+1}{4}\).

Area \(BCE=BOE-BOC=\frac{\sqrt{2}+1}{4}-\frac{1}{4}=\frac{\sqrt{2}}{4}\)

Answer: B.

Attachment:
Square ABCD.JPG







Hi Bunuel,

Please, can you explain this in lame terms as to how the area of BOC is one forth and the diagonal part?


Is there any other way to solve it?
Math Expert
User avatar
V
Joined: 02 Sep 2009
Posts: 58954
Re: In the figure, each side of square ABCD has length 1, the length of li  [#permalink]

Show Tags

New post 19 Feb 2019, 01:14
Shef08 wrote:
Bunuel wrote:
Image
In the figure, each side of square ABCD has length 1, the length of line Segment CE is 1, and the length of line segment BE is equal to the length of line segment DE. What is the area of the triangular region BCE?

A. 1/3

B. \(\frac{\sqrt{2}}{4}\)

C. 1/2

D. \(\frac{\sqrt{2}}{2}\)

E. 3/4


Image

Note that as BE=DE the triangles BCE and CDE are congruent (all three sides are equal) --> \(\angle{BCE}=\angle{DCE}\). So if we extend the line segment CE it will meet with the crosspoint of the diagonals, let's call it point O. Also note that EO will be perpendicular to BD, as diagonals in square make a right angle.

(The area of BCE) = (The area of BOE) - (The area of BOC).

Area of BOC is one fourth of the square's =\(\frac{1}{4}\).

Area BOE, \(\frac{1}{2}*BO*EO\). \(BO=\frac{\sqrt{2}}{2}\), half of the diagonal of a square. \(EO=CE+CO=1+\frac{\sqrt{2}}{2}=\frac{2+\sqrt{2}}{2}\), CO is also half of the diagonal of a square.

So \(AreaBOE=\frac{1}{2}*BO*EO=\frac{1}{2}*\frac{\sqrt{2}}{2}*\frac{2+\sqrt{2}}{2}=\frac{\sqrt{2}+1}{4}\).

Area \(BCE=BOE-BOC=\frac{\sqrt{2}+1}{4}-\frac{1}{4}=\frac{\sqrt{2}}{4}\)


Answer: B.

Attachment:
The attachment Square ABCD.JPG is no longer available







Hi Bunuel,

Please, can you explain this in lame terms as to how the area of BOC is one forth and the diagonal part?


Is there any other way to solve it?


Image
Diagonals of a square are perpendicular bisectors of each other. That is, they cut each other in half and at 90 degrees. As you can see above diagonals of a square cut the square into four equal parts.

For other solutions please check the discussion. There are many different solutions there.

Attachment:
Untitled.png
Untitled.png [ 27.17 KiB | Viewed 732 times ]

_________________
Manager
Manager
avatar
G
Joined: 31 Jul 2017
Posts: 87
Location: India
Schools: Anderson '21, LBS '21
GMAT ToolKit User Premium Member
In the figure, each side of square ABCD has length 1, the length of li  [#permalink]

Show Tags

New post 16 Mar 2019, 06:17
singh_amit19 wrote:
Image
In the figure, each side of square ABCD has length 1, the length of line Segment CE is 1, and the length of line segment BE is equal to the length of line segment DE. What is the area of the triangular region BCE?


A. 1/3

B. \(\frac{\sqrt{2}}{4}\)

C. 1/2

D. \(\frac{\sqrt{2}}{2}\)

E. 3/4


Attachment:
The attachment squareabcd.jpg is no longer available



Hi Bunuel,

I tried solving this problem in the following way.

\(\triangle\) BCD and \(\triangle\) CDE are similar triangles (Side-Side-Side)

So \(\angle\) DCE =\(\angle\) BCE

\(\angle\)DCE+\(\angle\)BCE=360-90= 270 degrees

So \(\angle\) DCE=\(\angle\)BCE =135 degree

Area of any triangle = 1/2 ab sin (angle between the sides)
= 1/2 *1 * 1 Sin(90+45)
=\(\frac{1}{2\sqrt[]{2}}\)
=\(\frac{\sqrt{2}}{4}\)

This way of solving would eliminate drawing extra sides in the figure. i hope you find this helpful.
Attachments

IMG_20190316_182755.jpg
IMG_20190316_182755.jpg [ 2.21 MiB | Viewed 613 times ]

Senior Manager
Senior Manager
User avatar
P
Status: Gathering chakra
Joined: 05 Feb 2018
Posts: 440
Premium Member
In the figure, each side of square ABCD has length 1, the length of li  [#permalink]

Show Tags

New post 23 May 2019, 17:39
~45 SECOND "SOLUTION"
I estimated the height of triangle BCE by using my pen after measuring the side square ABCE. I estimated BCE height to be 3/4 as ABCE side length was 1.
So b*h/2 ... (1*3/4) / 2 = 3/8 = .375

Looking at answers starting with C) which is .5, we need something smaller.
Trying B) √2/4 = 1.4/4 = .7/2 = .35
Trying A) 1/3 = .33
Between these two, B) is closer to .375 so I picked it.

It's not foolproof but it's the best method for Geometry when you're running low on time and not seeing a quick solution, imo.
VP
VP
User avatar
D
Joined: 14 Feb 2017
Posts: 1263
Location: Australia
Concentration: Technology, Strategy
GMAT 1: 560 Q41 V26
GMAT 2: 550 Q43 V23
GMAT 3: 650 Q47 V33
GMAT 4: 650 Q44 V36
GMAT 5: 650 Q48 V31
GPA: 3
WE: Management Consulting (Consulting)
Reviews Badge CAT Tests
Re: In the figure, each side of square ABCD has length 1, the length of li  [#permalink]

Show Tags

New post 22 Sep 2019, 19:44
Do we seriously always assume 2D figures????

Holy cow... i've only realised this now... this would have made my life so much easier!
_________________
Goal: Q49, V41

+1 Kudos if I have helped you
Intern
Intern
avatar
Joined: 21 Apr 2019
Posts: 1
Re: In the figure, each side of square ABCD has length 1, the length of li  [#permalink]

Show Tags

New post 09 Nov 2019, 22:25
Hey, I dont understand this part of Bunnel's explanation.

I too tried to find the area of the two triangles and subtract them. But I got lost in the calculations to find the actual height.

12∗BO∗EO12∗BO∗EO . BO=2√2
GMAT Club Bot
Re: In the figure, each side of square ABCD has length 1, the length of li   [#permalink] 09 Nov 2019, 22:25

Go to page   Previous    1   2   [ 29 posts ] 

Display posts from previous: Sort by

In the figure, each side of square ABCD has length 1, the length of li

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





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