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

 It is currently 15 Jul 2019, 11:48

### 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

# H,G,F and E are midpoints of the sides of square ABCD

Author Message
TAGS:

### Hide Tags

Intern
Joined: 18 Feb 2011
Posts: 38
GPA: 3.91
H,G,F and E are midpoints of the sides of square ABCD  [#permalink]

### Show Tags

11 Feb 2012, 00:48
1
00:00

Difficulty:

35% (medium)

Question Stats:

78% (02:16) correct 22% (02:29) wrong based on 286 sessions

### HideShow timer Statistics

H,G,F and E are midpoints of the sides of square ABCD. Arcs FG and EH are centered at B and D respectively, as shown above. If the side of the square ABCD is 4, what is the area of the shaded region HEFG?

A) 4(3-$$\pi$$)
B) 2(4-$$\pi$$)
C) 4(4-$$\pi$$)
D) 2(6-$$\pi$$)
E) 8(1+$$\pi$$)
Math Expert
Joined: 02 Sep 2009
Posts: 56226
Re: H,G,F and E are midpoints of the sides of square ABCD  [#permalink]

### Show Tags

11 Feb 2012, 01:14
1
nafishasan60 wrote:

H,G,F and E are midpoints of the sides of square ABCD. Arcs FG and EH are centered at B and D respectively, as shown above. If the side of the square ABCD is 4, what is the area of the shaded region HEFG?

A) 4(3-$$\pi$$)
B) 2(4-$$\pi$$)
C) 4(4-$$\pi$$)
D) 2(6-$$\pi$$)
E) 8(1+$$\pi$$)

The legs of right isosceles triangles FAE and GCH equal to half of the side of the square, so to 4/2=2, hence their combined area is $$2*(\frac{1}{2}*2*2)=4$$;

The same way the radii of arcs FG and EH equal to 4/2=2. Since both arcs are 90 degrees, then their commbined area is $$2*\frac{90}{360}*\pi*{r^2}=2\pi$$;

The are of the square is 4^2=16, thus the area of the shaded region is $$16-(4+2\p)=2(6-\pi)$$.

Hope it's clear.
_________________
Math Expert
Joined: 02 Sep 2009
Posts: 56226
Re: H,G,F and E are midpoints of the sides of square ABCD  [#permalink]

### Show Tags

06 Mar 2014, 02:34
Bumping for review and further discussion.

_________________
SVP
Status: The Best Or Nothing
Joined: 27 Dec 2012
Posts: 1787
Location: India
Concentration: General Management, Technology
WE: Information Technology (Computer Software)
Re: H,G,F and E are midpoints of the sides of square ABCD  [#permalink]

### Show Tags

07 Mar 2014, 01:15
Unshaded region has 2 equal quartercircles (means 1 semicircle) & 2 equal right triangles (Means 1 square)

Area of Semicircle = pie 2^2/2 = 2pi
Area of small Square = 4

Area of shaded region = 16 - 4 - 2pi

= 12 - 2pi
= 2(6-pi) = D
_________________
Kindly press "+1 Kudos" to appreciate
Retired Moderator
Joined: 29 Apr 2015
Posts: 830
Location: Switzerland
Concentration: Economics, Finance
Schools: LBS MIF '19
WE: Asset Management (Investment Banking)
H,G,F and E are midpoints of the sides of square ABCD  [#permalink]

### Show Tags

26 Sep 2015, 05:52
I would ballpark for the right solution. Divide the square into 4 small squares (blue line) and call the midpoint M. From this you see that the shaded part within AFME and MGCH is always half. From this you have each 1/8 = 2/8 of the total square. Lastly also divide EMHD and FBGM further into halfes (red lines). There you see that the shaded region is half of 1/8 and therefore 1/16. So add 2/16 to 2/8 to get 3/8.

Attachment:

Unbenannt.png [ 7.56 KiB | Viewed 4154 times ]

So we are looking for 3/8 of 16 = 6 in the answer choices. D is closest.
_________________
Saving was yesterday, heat up the gmatclub.forum's sentiment by spending KUDOS!

PS Please send me PM if I do not respond to your question within 24 hours.
Non-Human User
Joined: 09 Sep 2013
Posts: 11646
Re: H,G,F and E are midpoints of the sides of square ABCD  [#permalink]

### Show Tags

07 Apr 2017, 23:14
Hello from the GMAT Club BumpBot!

Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos).

Want to see all other topics I dig out? Follow me (click follow button on profile). You will receive a summary of all topics I bump in your profile area as well as via email.
_________________
Re: H,G,F and E are midpoints of the sides of square ABCD   [#permalink] 07 Apr 2017, 23:14
Display posts from previous: Sort by