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29 Sep 2021, 01:00
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Difficulty:
55%
(hard)
Question Stats:
51% (01:19) correct
49%
(01:10)
wrong
based on 196
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Is rectangle R a square?
(1) At least one side of rectangle R has an integer length.
(2) The diagonals of rectangle R have integer lengths.
(1) At least one side of rectangle R has an integer length.
(2) The diagonals of rectangle R have integer lengths.
20 Oct 2021, 07:39
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Is rectangle R a square?
So we have to check whether sides are equal.
(1) At least one side of rectangle R has an integer length.
Sides could be 4 each, then a square.
Sides could be 3,4,3,4, then a rectangle.
Insufficient
(2) The diagonals of rectangle R have integer lengths.
If diagonal is 4, the sides could be \(\frac{4}{\sqrt{2}}\)…Square
If diagonal is 5 and sides 3 and 4, then not a square.
Insufficient
Combined
The diagonal, say AC, and two sides, AB and BC, will make a right angled triangle.
If R is a square the ratio of sides AB:BC:AC are \(1:1:\sqrt{2}\).
That is, in a square, both the diagonal and two equal sides will never be integer as DIAGONAL is \(\sqrt{2}\) times the side.
Here, we have integer value for diagonal and a side, which will never be the case for a square.
Thus R is not a square.
Sufficient
C
So we have to check whether sides are equal.
(1) At least one side of rectangle R has an integer length.
Sides could be 4 each, then a square.
Sides could be 3,4,3,4, then a rectangle.
Insufficient
(2) The diagonals of rectangle R have integer lengths.
If diagonal is 4, the sides could be \(\frac{4}{\sqrt{2}}\)…Square
If diagonal is 5 and sides 3 and 4, then not a square.
Insufficient
Combined
The diagonal, say AC, and two sides, AB and BC, will make a right angled triangle.
If R is a square the ratio of sides AB:BC:AC are \(1:1:\sqrt{2}\).
That is, in a square, both the diagonal and two equal sides will never be integer as DIAGONAL is \(\sqrt{2}\) times the side.
Here, we have integer value for diagonal and a side, which will never be the case for a square.
Thus R is not a square.
Sufficient
C
General Discussion
