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Is quadrilateral ABCD a rectangle?

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Is quadrilateral ABCD a rectangle?  [#permalink]

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New post Updated on: 06 Jun 2015, 10:50
2
10
00:00
A
B
C
D
E

Difficulty:

  55% (hard)

Question Stats:

49% (01:07) correct 51% (00:54) wrong based on 230 sessions

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Is quadrilateral ABCD a rectangle?

(1) Two opposite sides of ABCD are equal in length.
(2) Two opposite internal angles of ABCD are right angles.

SOLUTION:
Stat.(1) only informs you about one pair of opposite sides. This can be true in rectangles and in other quadrilaterals, such as trapezoids (see figure below). Therefore, you cannot be sure that ABCD is a rectangle. Hence, Stat.(1)->Maybe->IS->BCE.
Attachment:
T6058a.png
T6058a.png [ 2.99 KiB | Viewed 2987 times ]


Stat.(2) only informs you about one pair of opposite opposites angles. As the other two angles can vary, this does not indicate whether ABCD is a rectangle. The figure below shows two quadrilaterals with two opposite right angles - one is a rectangle, the other isn't. Hence, Stat.(2)->Maybe->IS->CE.
Attachment:
T6058b.png
T6058b.png [ 3 KiB | Viewed 2987 times ]


Stat.(1+2) combined are a different story. Now you have two pairs of equal sides with opposite right angles. Try to draw it. Start with a rectangle, and play with the other two angles and sides (marked in red), trying to create a quadrilateral which isn't a rectangle:

[figure of two sides + right angles in various constellations]
Attachment:
T6058c.png
T6058c.png [ 5.68 KiB | Viewed 2984 times ]


The only possible way to create a quadrilateral is if the others angles are right too. ABCD must be a rectangle. Hence, Stat.(1+2)->Yes->S->C.

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Originally posted by reto on 06 Jun 2015, 04:17.
Last edited by Bunuel on 06 Jun 2015, 10:50, edited 3 times in total.
Renamed the topic and edited the question.
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Re: Is quadrilateral ABCD a rectangle?  [#permalink]

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New post 10 Jun 2015, 11:48
2
2
reto wrote:
Is quadrilateral ABCD a rectangle?

(1) Two opposite sides of ABCD are equal in length.
(2) Two opposite internal angles of ABCD are right angles.

SOLUTION:
Stat.(1) only informs you about one pair of opposite sides. This can be true in rectangles and in other quadrilaterals, such as trapezoids (see figure below). Therefore, you cannot be sure that ABCD is a rectangle. Hence, Stat.(1)->Maybe->IS->BCE.
Attachment:
T6058a.png


Stat.(2) only informs you about one pair of opposite opposites angles. As the other two angles can vary, this does not indicate whether ABCD is a rectangle. The figure below shows two quadrilaterals with two opposite right angles - one is a rectangle, the other isn't. Hence, Stat.(2)->Maybe->IS->CE.
Attachment:
T6058b.png


Stat.(1+2) combined are a different story. Now you have two pairs of equal sides with opposite right angles. Try to draw it. Start with a rectangle, and play with the other two angles and sides (marked in red), trying to create a quadrilateral which isn't a rectangle:

[figure of two sides + right angles in various constellations]
Attachment:
T6058c.png


The only possible way to create a quadrilateral is if the others angles are right too. ABCD must be a rectangle. Hence, Stat.(1+2)->Yes->S->C.


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Re: Is quadrilateral ABCD a rectangle?  [#permalink]

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New post 09 Jul 2016, 05:50
I don't get one thing here. When we say "rectangle" do we mean "a square"? I mean every square is a rectangle but not every rectangle is a square, right? In this problem rectangle can be a square and still answer is correct, right?
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Re: Is quadrilateral ABCD a rectangle?  [#permalink]

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New post 01 Sep 2016, 09:00
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What if it's a square? Is the assumption that every 'square' is a rectangle?
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Re: Is quadrilateral ABCD a rectangle?  [#permalink]

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New post 01 Sep 2016, 09:27
dina98 wrote:
What if it's a square? Is the assumption that every 'square' is a rectangle?



Hi,
Yes, every square is a rectangle because it fulfills all requirements of a rectangle like opposite sides equal, diagonal equal and all angles are 90...
It is just that additionally a square has not only OPP sides but all sides are equal..
And the diag are not only equal and bisect each other but also cut each other at 90..

Thus all square are rectangle but all rectangle need not be square...

Hope it helps
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Re: Is quadrilateral ABCD a rectangle?  [#permalink]

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New post 08 Jul 2018, 12:37
chetan2u wrote:
dina98 wrote:
What if it's a square? Is the assumption that every 'square' is a rectangle?



Hi,
Yes, every square is a rectangle because it fulfills all requirements of a rectangle like opposite sides equal, diagonal equal and all angles are 90...
It is just that additionally a square has not only OPP sides but all sides are equal..
And the diag are not only equal and bisect each other but also cut each other at 90..

Thus all square are rectangle but all rectangle need not be square...

Hope it helps

what is the difference?
mattce wrote:

Is quadrilateral ABCD a rectangle?

(1) Two sides of ABCD are equal in length

(2) Two opposite internal angles of ABCD are right angles


Thought this was a good question; really tested my visualization. (Please don't think harder just because I said that...)


The question has quite ambiguous wording...

Anyway, check the diagram below:
Attachment:

E.png
E.png [ 2.77 KiB | Viewed 4395 times ]

Answer: E.
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Re: Is quadrilateral ABCD a rectangle?  [#permalink]

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New post 27 Jul 2019, 06:10
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Re: Is quadrilateral ABCD a rectangle?   [#permalink] 27 Jul 2019, 06:10
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