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# If the vertices of a quadrilateral PQRS lie on the circumference of

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If the vertices of a quadrilateral PQRS lie on the circumference of  [#permalink]

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Updated on: 11 Nov 2017, 04:13
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Difficulty:

95% (hard)

Question Stats:

41% (01:46) correct 59% (01:59) wrong based on 264 sessions

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If the vertices of a quadrilateral PQRS lie on the circumference of the circle, is PQRS a square?

(1) Side PS is equal in length to the radius of the circle
(2) The degree measure of the minor arc QR is 90 degrees

Originally posted by joylive on 16 Oct 2012, 07:35.
Last edited by Bunuel on 11 Nov 2017, 04:13, edited 2 times in total.
Edited the OA.
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Re: If the vertices of a quadrilateral PQRS lie on the circumference of  [#permalink]

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16 Oct 2012, 08:16
5
1
joylive wrote:
If the vertices of a quadrilateral PQRS lie on the circumference of the circle, is PQRS a square?
1. Side PS is equal in length to the radius of the circle
2. The degree measure of the minor arc QR is 90 degrees

Can you explain this?

Would like to know what is the source of this problem and how OA is D? It should be solved as below:

First consider this,
if a square has all of its veritces on circumference of the circle and if radius of circle is r and side of square is x. then $$x= r*\sqrt{2}$$
This can be proved using pyth theorem.
Second diagonals are perpendicular to each other.

Check the image screenshot 1

Now lets go to statements 1 by 1.

Statement 1 says:
PS = radius (where PS is one of the sides)
Or basically x=r
however we know for a square $$x= r*\sqrt{2}$$
another way to look at same is - if side is equal to radius - all angles are 60 in the triangle POS. However, for a square angle POS should be 90.
Hence, we know that PQRS is not a square, as it doesnt meet even the basic requirement.

Hence statement is Sufficient

Statement 2 says:
QOR =90 Where O is center of circle.
Hence, using pyth theorem in triangle QOR,
QR^2 = OR^2 + OQ^2
or
x^2 = r^2 + r^2
or $$x= r*\sqrt{2}$$
that means, side length is fine. side QS meets minimum condition, but not sufficient to answer anything else since we dont know anything about rest of the sides or angles, it could be a quad as shown in image 2.

Hence statement 2 is not sufficient.

Ans is A.
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Screenshot01 17-Oct-12 00.12.jpg [ 15.21 KiB | Viewed 7049 times ]

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Re: If the vertices of a quadrilateral PQRS lie on the circumference of  [#permalink]

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16 Oct 2012, 08:40
joylive wrote:
If the vertices of a quadrilateral PQRS lie on the circumference of the circle, is PQRS a square?
1. Side PS is equal in length to the radius of the circle
2. The degree measure of the minor arc QR is 90 degrees

Can you explain this?

(1) The side of a square inscribed in a given circle is $$R\sqrt{2}$$, where $$R$$ is the radius of the circle.
Since PS = R, the quadrilateral PQRS cannot be a square.
Sufficient.

(2) The line segment QR subtends an arc of 90 degrees, therefore QR equals the side of the square which can be inscribed in the given circle.
In an inscribed square, each side subtends an arc of 360/4 = 90 degrees.
Since we have no information about the other sides of the quadrilateral, PQRS isn't necessarily a square.
Not sufficient.

Answer A.
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Re: If the vertices of a quadrilateral PQRS lie on the circumference of  [#permalink]

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27 Jul 2013, 04:23
EvaJager wrote:
joylive wrote:
If the vertices of a quadrilateral PQRS lie on the circumference of the circle, is PQRS a square?
1. Side PS is equal in length to the radius of the circle
2. The degree measure of the minor arc QR is 90 degrees

Can you explain this?

(1) The side of a square inscribed in a given circle is $$R\sqrt{2}$$, where $$R$$ is the radius of the circle.
Since PS = R, the quadrilateral PQRS cannot be a square.
Sufficient.

(2) The line segment QR subtends an arc of 90 degrees, therefore QR equals the side of the square which can be inscribed in the given circle.
In an inscribed square, each side subtends an arc of 360/4 = 90 degrees.
Since we have no information about the other sides of the quadrilateral, PQRS isn't necessarily a square.
Not sufficient.

Answer A.

Thanks for the answer,

1) However, what if the statement 2 read - minor arc as 60 ?
In that case Is it sufficient to answer the question ?

2) Also, in case when the minor arc is 90,
Can't we use the vertical angles concept and prove that other angles too are 90 ? and subsequently 360 for a circular angle ?
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Re: If the vertices of a quadrilateral PQRS lie on the circumference of  [#permalink]

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29 Mar 2014, 14:21
1
OA says E. Is there any good reason for this?

Thanks
Cheers
J
Math Expert
Joined: 02 Sep 2009
Posts: 52344
Re: If the vertices of a quadrilateral PQRS lie on the circumference of  [#permalink]

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29 Mar 2014, 14:28
jlgdr wrote:
OA says E. Is there any good reason for this?

Thanks
Cheers
J

The correct answer must be A, not E. Edited the OA. Thank you for reporting this.
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Re: If the vertices of a quadrilateral PQRS lie on the circumference of  [#permalink]

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25 Dec 2014, 02:49
option A states that radius is equal to one side this itself confirms that it cant be a square as for square it has to be rt 2 times the radius.
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Re: If the vertices of a quadrilateral PQRS lie on the circumference of  [#permalink]

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04 Jul 2018, 07:07
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Can someone throw more light on the second statement?
I get the first, but how do we derive the value of the angle of a quadrilateral, from the arc of a circle that it is circumscribed by?

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Re: If the vertices of a quadrilateral PQRS lie on the circumference of &nbs [#permalink] 04 Jul 2018, 07:07
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# If the vertices of a quadrilateral PQRS lie on the circumference of

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