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# Is quadrilateral ABCD a rectangle?

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Updated on: 06 Jun 2015, 10:50
2
14
00:00

Difficulty:

55% (hard)

Question Stats:

47% (01:05) correct 53% (00:53) wrong based on 289 sessions

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(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 [ 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 [ 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 [ 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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10 Jun 2015, 11:48
4
2
reto wrote:

(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.

Similar questions to practice:
if-a-b-and-c-are-distinct-points-135308.html
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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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01 Sep 2016, 09:00
1
What if it's a square? Is the assumption that every 'square' is a rectangle?
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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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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:

(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 ]

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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

Posted from my mobile device
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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:

(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 ]

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18 Aug 2019, 21:24
Bunuel
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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18 Aug 2019, 21:36
1
globaldesi wrote:
Bunuel
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:

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18 Aug 2019, 21:44
Bunuel wrote:
globaldesi wrote:
Bunuel
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:

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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18 Aug 2019, 21:46
1
globaldesi wrote:
Bunuel wrote:
globaldesi wrote:
Bunuel
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:

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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18 Aug 2019, 21:48
globaldesi wrote:
Bunuel wrote:
globaldesi wrote:
Bunuel
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:

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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