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In the figure given below, ABCD is a square, and P, Q, R and

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b2bt
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In the figure given below, ABCD is a square, and P, Q, R and [#permalink] New post   03 Oct 2013, 02:12
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68% (02:12) correct 33% (02:05) wrong based on 600 sessions
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tS6VTJ2.jpg
tS6VTJ2.jpg [ 11.04 KiB | Viewed 32353 times ]

In the figure given below, ABCD is a square, and P, Q, R and S are the mid-points of the sides AB, BC, CD and AD respectively. The ratio of the area of the shaded region to the area of the square ABCD is

A. 1/3
B. 1/4
C. 1/5
D. 1/6
E. 1/8
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B
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Re: In the figure given below, ABCD is a square, and P, Q, R and [#permalink] New post   03 Oct 2013, 02:27
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b2bt wrote:
Attachment:
The attachment tS6VTJ2.jpg is no longer available

In the figure given below, ABCD is a square, and P, Q, R and S are the mid-points of the sides AB, BC, CD and AD respectively. The ratio of the area of the shaded region to the area of the square ABCD is

A. 1/3
B. 1/4
C. 1/5
D. 1/6
E. 1/8


Attachment:
Untitled.png
Untitled.png [ 30.11 KiB | Viewed 32227 times ]

Consider square APNS and say its side is 1. In this case:

The area of APNS is 1.
MN = 1/2, which means that the area of SMN is 1/2*1/2*1=1/4.

(shaded)/(square)=(1/4)/1=1/4.

The ratio for the entire square would be the same.

Answer: B.
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Re: In the figure given below, ABCD is a square, and P, Q, R and [#permalink] New post   03 Oct 2013, 03:08
Bunuel wrote:
b2bt wrote:
Attachment:
tS6VTJ2.jpg

In the figure given below, ABCD is a square, and P, Q, R and S are the mid-points of the sides AB, BC, CD and AD respectively. The ratio of the area of the shaded region to the area of the square ABCD is

A. 1/3
B. 1/4
C. 1/5
D. 1/6
E. 1/8


Attachment:
Untitled.png

Consider square APNS and say its side is 1. In this case:

The area of APNS is 1.
MN = 1/2, which means that the area of SMN is 1/2*1/2*1=1/4.

(shaded)/(square)=(1/4)/1=1/4.

The ratio for the entire square would be the same.

Answer: B.



How did you get MN as half? I solved the same way but assumed it to half...
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Re: In the figure given below, ABCD is a square, and P, Q, R and [#permalink] New post   18 Nov 2013, 19:39
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APNS is a square. It's side is 1(as assumed by buneul). This means that ABQS is a rectangle with AS=1 and SQ=2. Hence, PN=1. SB and AQ are diagonals of the rectangle. Let them meet at the point M. Draw an imaginary line say XY passing through M and parallel to both AB and SQ. Now observe that PN bisects SQ that is SN=NQ=1. This implies that XY passing through M should also bisect AS, PN and BQ. Therefore, MN=1/2. This property holds good for all rectangles.
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Re: In the figure given below, ABCD is a square, and P, Q, R and [#permalink] New post   19 Nov 2013, 04:17
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b2bt wrote:
Bunuel wrote:
b2bt wrote:
Attachment:
tS6VTJ2.jpg

In the figure given below, ABCD is a square, and P, Q, R and S are the mid-points of the sides AB, BC, CD and AD respectively. The ratio of the area of the shaded region to the area of the square ABCD is

A. 1/3
B. 1/4
C. 1/5
D. 1/6
E. 1/8


Attachment:
Untitled.png

Consider square APNS and say its side is 1. In this case:

The area of APNS is 1.
MN = 1/2, which means that the area of SMN is 1/2*1/2*1=1/4.


(shaded)/(square)=(1/4)/1=1/4.

The ratio for the entire square would be the same.

Answer: B.



How did you get MN as half? I solved the same way but assumed it to half...



Hello B2bt,

Consider triangle SMN and SBQ

Angles S is common
Angle N is equal to Angle Q and
Angle M is equal to angle B

Since the triangles are similar therefore MN=1/2BQ ie. 1/2
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Re: In the figure given below, ABCD is a square, and P, Q, R and [#permalink] New post   12 Sep 2016, 10:09
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b2bt wrote:
Attachment:
tS6VTJ2.jpg

In the figure given below, ABCD is a square, and P, Q, R and S are the mid-points of the sides AB, BC, CD and AD respectively. The ratio of the area of the shaded region to the area of the square ABCD is

A. 1/3
B. 1/4
C. 1/5
D. 1/6
E. 1/8


The figure can be seen as four identical squares with identical shaded region and each individual square. And the shaded region is from one end to the mid point of the other corner. This means it's 1/4 of the area of the main square. Since these are identical, the Answer is B) 1/4
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In the figure given below, ABCD is a square, and P, Q, R and [#permalink] New post   Updated on: 21 Sep 2018, 06:25
Since the answer choices are quite far from one another, the best way for such graphical problems is approximation.
We have got 4 right triangles, which you can easily fit into one of the small triangle (1/4 of original one). Hence the ratio is 1:4. Time taken 47 sec.

Originally posted by Hero8888 on 22 Aug 2018, 07:18.
Last edited by Hero8888 on 21 Sep 2018, 06:25, edited 1 time in total.
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Re: In the figure given below, ABCD is a square, and P, Q, R and [#permalink] New post   20 Sep 2018, 21:54
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b2bt wrote:
Attachment:
The attachment tS6VTJ2.jpg is no longer available

In the figure given below, ABCD is a square, and P, Q, R and S are the mid-points of the sides AB, BC, CD and AD respectively. The ratio of the area of the shaded region to the area of the square ABCD is

A. 1/3
B. 1/4
C. 1/5
D. 1/6
E. 1/8


In such question Most important part is to break the shaded portion into a few recognizable figures like I have broken as shown in figure attached

Answer: Option B
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Screen Shot 2018-09-21 at 10.16.34 AM.png
Screen Shot 2018-09-21 at 10.16.34 AM.png [ 359.56 KiB | Viewed 14938 times ]

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Re: In the figure given below, ABCD is a square, and P, Q, R and [#permalink] New post   17 Feb 2020, 22:36
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b2bt wrote:
Attachment:
tS6VTJ2.jpg

In the figure given below, ABCD is a square, and P, Q, R and S are the mid-points of the sides AB, BC, CD and AD respectively. The ratio of the area of the shaded region to the area of the square ABCD is

A. 1/3
B. 1/4
C. 1/5
D. 1/6
E. 1/8



Adding Video solution to the thread.

Answer: Option B

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Re: In the figure given below, ABCD is a square, and P, Q, R and [#permalink] New post   14 Mar 2021, 23:39
Am I guilty? It is easily recognizable that the shaded region makes up 1/4 of the square. Each smaller triangle makes up 1/4 of 1/4 of the square.

Is there a better analysis?
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Re: In the figure given below, ABCD is a square, and P, Q, R and [#permalink] New post   11 Feb 2023, 13:17
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Re: In the figure given below, ABCD is a square, and P, Q, R and [#permalink]
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