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Is quadrilateral PQRS a parallelogram? (1) P, Q, R, S are the mid-poi

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Is quadrilateral PQRS a parallelogram? (1) P, Q, R, S are the mid-poi  [#permalink]

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New post 25 Jun 2018, 11:06
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Is quadrilateral PQRS a parallelogram?

(1) P, Q, R, S are the mid-points of sides AB, BC, CD and AD respectively of a trapezoid ABCD.

(2) Diagonals of quadrilateral PQRS bisect each other.
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Re: Is quadrilateral PQRS a parallelogram? (1) P, Q, R, S are the mid-poi  [#permalink]

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New post 28 Jun 2018, 07:12
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amanvermagmat wrote:
Is quadrilateral PQRS a parallelogram?

(1) P, Q, R, S are the mid-points of sides AB, BC, CD and AD respectively of a trapezoid ABCD.

(2) Diagonals of quadrilateral PQRS bisect each other.



With 90% wrong, it shows lack of properties of a parallelogram....

(1) P, Q, R, S are the mid-points of sides AB, BC, CD and AD respectively of a trapezoid ABCD.
When midpoints are joined of the four sides of ANY quadrilateral, it forms a parallelogram
look at the sketch..
ABCD is a quadrilateral and PQRS forms another quadrilateral by joining the midpoints..
join diagonal DB....
in triangle ADB, PQ will be parallel and half of the diagonal DB as PQ is bisecting the other two sides..
similarly SR is also parallel and half of DB..
thus PQ||DB||SR and PQ=SR

similarly for the set of other opposite sides
hence the quadrilateral is parallelogram

sufficient

(2) Diagonals of quadrilateral PQRS bisect each other.
Again this is the property of a quadrilateral- if diagonals bisect each other, it is a parallelogram
ca be proven by similar triangles and congruency..
sufficient

D
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Re: Is quadrilateral PQRS a parallelogram? (1) P, Q, R, S are the mid-poi  [#permalink]

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New post 28 Jun 2018, 08:17
chetan2u wrote:
amanvermagmat wrote:
Is quadrilateral PQRS a parallelogram?

(1) P, Q, R, S are the mid-points of sides AB, BC, CD and AD respectively of a trapezoid ABCD.

(2) Diagonals of quadrilateral PQRS bisect each other.



With 90% wrong, it shows lack of properties of a parallelogram....

(1) P, Q, R, S are the mid-points of sides AB, BC, CD and AD respectively of a trapezoid ABCD.
When midpoints are joined of the four sides of ANY quadrilateral, it forms a parallelogram
look at the sketch..
ABCD is a quadrilateral and PQRS forms another quadrilateral by joining the midpoints..
join diagonal DB....
in triangle ADB, PQ will be parallel and half of the diagonal DB as PQ is bisecting the other two sides..
similarly SR is also parallel and half of DB..
thus PQ||DB||SR and PQ=SR

similarly for the set of other opposite sides
hence the quadrilateral is parallelogram

sufficient

(2) Diagonals of quadrilateral PQRS bisect each other.
Again this is the property of a quadrilateral- if diagonals bisect each other, it is a parallelogram
ca be proven by similar triangles and congruency..
sufficient

D


Hi chetan2u

In statement 2, Could the figure be square or rectangular? or are both subset/special case of parallelogram so they must be considered parallelogram?

Thanks
Math Expert
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Joined: 02 Aug 2009
Posts: 6815
Re: Is quadrilateral PQRS a parallelogram? (1) P, Q, R, S are the mid-poi  [#permalink]

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New post 28 Jun 2018, 08:51
Mo2men wrote:
chetan2u wrote:
amanvermagmat wrote:
Is quadrilateral PQRS a parallelogram?

(1) P, Q, R, S are the mid-points of sides AB, BC, CD and AD respectively of a trapezoid ABCD.

(2) Diagonals of quadrilateral PQRS bisect each other.



With 90% wrong, it shows lack of properties of a parallelogram....

(1) P, Q, R, S are the mid-points of sides AB, BC, CD and AD respectively of a trapezoid ABCD.
When midpoints are joined of the four sides of ANY quadrilateral, it forms a parallelogram
look at the sketch..
ABCD is a quadrilateral and PQRS forms another quadrilateral by joining the midpoints..
join diagonal DB....
in triangle ADB, PQ will be parallel and half of the diagonal DB as PQ is bisecting the other two sides..
similarly SR is also parallel and half of DB..
thus PQ||DB||SR and PQ=SR

similarly for the set of other opposite sides
hence the quadrilateral is parallelogram

sufficient

(2) Diagonals of quadrilateral PQRS bisect each other.
Again this is the property of a quadrilateral- if diagonals bisect each other, it is a parallelogram
ca be proven by similar triangles and congruency..
sufficient

D


Hi chetan2u

In statement 2, Could the figure be square or rectangular? or are both subset/special case of parallelogram so they must be considered parallelogram?

Thanks



Hi..

It can be rectangle or square..
Figure with four sides- quadrilateral
Parallelogram is a type of quadrilateral with opposite sides equal and parallel.
Rectangle is a type of parallelogram and each square is a rectangle..
Rhombus is a parallelogram and each square is also a rhombus
Apart from a parallelogram, trapezium is another type of quadrilateral
_________________

1) Absolute modulus : http://gmatclub.com/forum/absolute-modulus-a-better-understanding-210849.html#p1622372
2)Combination of similar and dissimilar things : http://gmatclub.com/forum/topic215915.html
3) effects of arithmetic operations : https://gmatclub.com/forum/effects-of-arithmetic-operations-on-fractions-269413.html


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Re: Is quadrilateral PQRS a parallelogram? (1) P, Q, R, S are the mid-poi &nbs [#permalink] 28 Jun 2018, 08:51
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