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11 Dec 2014, 07:21
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A
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Difficulty:
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Question Stats:
78% (01:15) correct
22%
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wrong
based on 357
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The figure above shows four pieces of tile that have been glued together to form a square tile ABCD. Is PR=QS?
(1) BQ = CR = DS = AP
(2) The perimeter of ABCD is 16.
Attachment:
2014-12-11_1820.png [ 4.3 KiB | Viewed 9867 times ]
01 Aug 2017, 04:16
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Really tricky question,
Initially marked answer E, as got all sides of quadrilateral formed inside equal but have no detail about diagonals.
But after read that correct option is A, came up with below solution. Seems it was an easy one.
Easy and simple steps, but to explain, i have written everything in detail.
Attached the hand drawn image.
Solution:
Given Final image is a square. SO all sides are equal
Taking point 1:
BQ=CR=DS=AP = a (suppose)
let remaining portion of side = b
So total square side =a+b
Draw perpendicular from Q on AD => QE . So QE = side of square= a+b
same way Draw perpendicular from P on CD => PF . So PF = side of square= a+b
Now, as its square, all vertices are 90 degree. As BE perpendicular on AD => BQ=AE =a and BA=QE= a+b
As AD= a+b and now AE=a and given SD=a there fore ES = a+b -a -a = b-a
So right triangle triangle QES has 1 side QE=a+b and 2nd side ES=b-a. So we can find hypotenuse QS .................(1)
Same way PFR makes a right triangle with 1 side PF= a+b and 2nd side RF= b-a. So we can find hypotenuse PR ...................(2)
By above 2 equations (1) and (2),
both triangles have same sides so hypotenuse length will be same
There for PR = QS
So option 1 is sufficient.
Taking point 2:
Given side of square =16. Doesn't give us any information
So option 2 not sufficient
Answer = A
Hope above explanation makes sense.
Initially marked answer E, as got all sides of quadrilateral formed inside equal but have no detail about diagonals.
But after read that correct option is A, came up with below solution. Seems it was an easy one.
Easy and simple steps, but to explain, i have written everything in detail.
Attached the hand drawn image.
Solution:
Given Final image is a square. SO all sides are equal
Taking point 1:
BQ=CR=DS=AP = a (suppose)
let remaining portion of side = b
So total square side =a+b
Draw perpendicular from Q on AD => QE . So QE = side of square= a+b
same way Draw perpendicular from P on CD => PF . So PF = side of square= a+b
Now, as its square, all vertices are 90 degree. As BE perpendicular on AD => BQ=AE =a and BA=QE= a+b
As AD= a+b and now AE=a and given SD=a there fore ES = a+b -a -a = b-a
So right triangle triangle QES has 1 side QE=a+b and 2nd side ES=b-a. So we can find hypotenuse QS .................(1)
Same way PFR makes a right triangle with 1 side PF= a+b and 2nd side RF= b-a. So we can find hypotenuse PR ...................(2)
By above 2 equations (1) and (2),
both triangles have same sides so hypotenuse length will be same
There for PR = QS
So option 1 is sufficient.
Taking point 2:
Given side of square =16. Doesn't give us any information
So option 2 not sufficient
Answer = A
Hope above explanation makes sense.
Attachments
Geomtery question.jpg [ 1.22 MiB | Viewed 7758 times ]
General Discussion
11 Dec 2014, 09:40
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In the Figure, From option A if we consider BQ=CR=DS=AP and project their mirror images on the opposite sides, then we would obtain four isosclese Right angled triangles at the four vertices. Now, further, for PR and QS , their perpendicular lengths are equal , and since we have projected the given four lengths on their opposite sides, therefore the BASES of the two triangles are also equal (the triangles which are formed by taking PR and OS as their hypotenuse and whose base is obtained by drawing a perpendicular side from the top vertex on the opposite side) .
Hence , since base and perpendicular lengths of the two triangles are equal, so that the sides PR and QS are equal in length taking Statement A as true.
And Statement B is clearly insufficient.
option A is the appropriate choice.
Hence , since base and perpendicular lengths of the two triangles are equal, so that the sides PR and QS are equal in length taking Statement A as true.
And Statement B is clearly insufficient.
option A is the appropriate choice. 
