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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
14
00:00
A
B
C
D
E

Difficulty:

  55% (hard)

Question Stats:

47% (01:05) correct 53% (00:53) wrong based on 289 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 3549 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 3578 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 3546 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.

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
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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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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 18 Aug 2019, 14:35
Can someone explain why is b wrong.

Can someone please draw a quadrilateral where opposite internal angles are 90 degrees but is not a rectangle

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

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New post 18 Aug 2019, 17:50
I couldn't find the invisible drawing of a quadrilateral having two opposite right angle and that quadrilateral is not rectangle.

I have tried many ways but I got only square and rectangle with condition B.
And each square is a rectangle.

I think B should be answer.
vivapopo wrote:
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 18 Aug 2019, 21:24
Bunuel
Can you please explain why B is not an answer?
If 2 sides are 90 degress that are opposite. It should be a rectangle. Could not imagine a figure which is not rectangle
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Re: Is quadrilateral ABCD a rectangle?  [#permalink]

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New post 18 Aug 2019, 21:36
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Re: Is quadrilateral ABCD a rectangle?  [#permalink]

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New post 18 Aug 2019, 21:44
Bunuel wrote:
globaldesi wrote:
Bunuel
Can you please explain why B is not an answer?
If 2 sides are 90 degress that are opposite. It should be a rectangle. Could not imagine a figure which is not rectangle


You could check under the spoiler in the original post to see the solution:
Image


Bunuel but the second figure you posted is not a quadrilateral.
Also when we say its a rectangle , do we eliminate an option that a square is also a rectangle
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Re: Is quadrilateral ABCD a rectangle?  [#permalink]

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New post 18 Aug 2019, 21:46
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globaldesi wrote:
Bunuel wrote:
globaldesi wrote:
Bunuel
Can you please explain why B is not an answer?
If 2 sides are 90 degress that are opposite. It should be a rectangle. Could not imagine a figure which is not rectangle


You could check under the spoiler in the original post to see the solution:
Image


Bunuel but the second figure you posted is not a quadrilateral.
Also when we say its a rectangle , do we eliminate an option that a square is also a rectangle


Why not? Does it have four sides? Yes! So, it's quadrilateral.

Your second question is not clear. All squares are rectangles but not all rectangles are squares.
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Re: Is quadrilateral ABCD a rectangle?  [#permalink]

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New post 18 Aug 2019, 21:48
globaldesi wrote:
Bunuel wrote:
globaldesi wrote:
Bunuel
Can you please explain why B is not an answer?
If 2 sides are 90 degress that are opposite. It should be a rectangle. Could not imagine a figure which is not rectangle


You could check under the spoiler in the original post to see the solution:
Image


Bunuel but the second figure you posted is not a quadrilateral.
Also when we say its a rectangle , do we eliminate an option that a square is also a rectangle


Why not? Does it have four sides? Yes! So, it's quadrilateral.

Your second question is not clear. All squares are rectangles but not all rectangles are squares.
Apologies.. i misinterpreted the figure.

Thanks the second line "All squares are rectangles but not all rectangles are squares" makes it clear
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Re: Is quadrilateral ABCD a rectangle?   [#permalink] 18 Aug 2019, 21:48
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