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# What is the perimeter of PQRS ? (1) x = 30 degree

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Senior Manager
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Re: What is the perimeter of PQRS ? (1) x = 30 degree  [#permalink]

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09 Dec 2013, 17:20
I think I understand, but not 100% positive that I got the correct answer.

The first hurdle to overcome is that I don't need a numeric answer? If that's the case, we're just looking to ensure that there are no possible different segment measurements given the line and angle measurements we have.

(1) x = 30 degree

We know x=30 and angle QTP = 60 so angle QPT = 90. Therefore, we know this is a right triangle and we can find the measurement of line QP.

The second thing to contend with is the image itself. All we know is that this is a quadrilateral and that for all we know, line RS may extend far more vertically that QT.

With that in mind, we know that angle QTS = 120 and that angle QRS = 120. However, we know nothing about angle RQS and angle QRS. Depending on their measurement, they may create different measurements for lines RS and QR while still maintaining QRS's 120 degree measure. The only thing we know about these two angles is that they must increase and decrease proportionately with one another/ We can figure out the measure of QP, PT, TS but not the measure of RS and QR. Insufficient.

(2) w= 45 degree

Knowing that QRS is 120 and QSR = 45, we know that SQR = 15. This is important because it fixes the lengths of the triangle so that they cannot move and change length. We are given the lengths of ST and PT which means that all we need to find is QP. While we only know one measure of triangle QTP, we know two of the side lengths and the angle opposite side QP is 60 which means lengths QT and QP are fixed and thus, the length of QP cannot change. Sufficient.

Does this make sense?
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Re: What is the perimeter of PQRS ? (1) x = 30 degree  [#permalink]

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31 Dec 2013, 10:29
Therefore the condition would be:

If we have a triangle which sides have respectively lenghts of: 1 , 2 and X . And if one of the angles is 60 than it is a right triangle with the ratio 1:2: root of 3

Am i right here?
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Re: What is the perimeter of PQRS ? (1) x = 30 degree  [#permalink]

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02 Jan 2014, 21:45
1
Paris75 wrote:
Therefore the condition would be:

If we have a triangle which sides have respectively lenghts of: 1 , 2 and X . And if one of the angles is 60 than it is a right triangle with the ratio 1:2: root of 3

Am i right here?

No. A triangle is uniquely defined by 2 sides and the INCLUDED angle. Not any angle. If two sides are 1 and 2, the triangles will be different depending on which angle is 60 degrees. If the angle between the sides of 1 and 2 length is 60, it will be a right triangle. Else it will not be right triangle.
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Re: What is the perimeter of PQRS ? (1) x = 30 degree  [#permalink]

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03 Jan 2014, 01:40
Is right hand side is not a parallelogram? in which opposite sides and opposite angels are equal?
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Re: What is the perimeter of PQRS ? (1) x = 30 degree  [#permalink]

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10 Jan 2014, 17:53
ronr34 wrote:
blueseas wrote:
Asifpirlo wrote:
What is the perimeter of PQRS ?
(1) x = 30 degree
(2) w= 45 degree

need some alternative ways

IN DS we dont need to calculate the exact value only we should make sure that answer is definite.

now in this question perimeter of PQRS is asked.
since this is quadrilateral hence sum of interior angles will be 360

statement 1:
$$x= 30$$
triangle QPT right triangle with angles 90 60 30...hence we can find out PQ
ANGLE Y = 120
Now if you see quad QRST ==>angle W and angle (Q-X) are not fixed...so to make angle R=120 we can make different figures. and hence there will be different perimeters.
hence insufficient.

statement 2:
w=45 now as quad QRST ==>sum of interior angle = 360
hence angle Q-X= 75
now since we have 2 sides fixed(QT and ST) and four angles are fixed...hence we are going to get a unique quad.hence we can calculate the length of QR and RS.

Now we have to find out only the length of PQ to determine the perimeter of quad PQRS.
SUM of interior angles of quad PQRS will be 360 hence ...P + W +120+Q =360
Now since Q= 75+X AND W=45
Putting the values and solving ...we get P+X = 120

NOW since P+X =120
One of the possibility is P=90 X=30 Hence in that case triangle PQT is right triangle with angle 90:60:30...hence side will be in ratio $$2:\sqrt{3}:1$$
now this clearly satisfies that triangle PQT is 90/60/30 TRIANGLE hence SIDE PQ = $$\sqrt{3}$$

HENCE we can find out the perimeter.hence sufficient

How do we know this? "hence angle Q-X= 75"

I too have the same doubt...
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Re: What is the perimeter of PQRS ? (1) x = 30 degree  [#permalink]

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12 Jan 2014, 19:42
1
lindt123 wrote:
I too have the same doubt...

Note that Q - X is just the angle TQR.
Angle Y is 180 -60 = 120 degrees.

All angles of the quadrilateral QRST add up to 360 = TQR + 120 + 45 + 120
So TQR = 75 degrees = Q - X
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Re: What is the perimeter of PQRS ? (1) x = 30 degree  [#permalink]

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26 Aug 2014, 09:50
I used the assumption from begining that QR and PS are paralell (looking at the diagram), and got to A .... but reading the posts made it clear that it was a false assumption. Is there any situation we can trust the illustration or it is there only to mislead? or a fact like that should be always stated?
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Re: What is the perimeter of PQRS ? (1) x = 30 degree  [#permalink]

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26 Aug 2014, 20:02
bigzoo wrote:
I used the assumption from begining that QR and PS are paralell (looking at the diagram), and got to A .... but reading the posts made it clear that it was a false assumption. Is there any situation we can trust the illustration or it is there only to mislead? or a fact like that should be always stated?

Do not draw inferences based on the diagrams. Perhaps the parallel lines have a slight slant that you are unable to make out. Perhaps the angle that seems 90 is actually 95 degrees you can't say. Also, the figures are to scale given the information in the question stem. They may not be to scale given the information in the 2 statements.
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Re: What is the perimeter of PQRS ? (1) x = 30 degree  [#permalink]

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26 Aug 2015, 07:19
VeritasPrepKarishma

If in a quadrilateral if one pair of opposite angles are equal then can we conclude that it will be a parallelogram?
In this example Y = 120 and QRS = 120. Can we conclude that QRST is a parallelogram?
Attachment:

SASASA.png [ 13.43 KiB | Viewed 1580 times ]

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Re: What is the perimeter of PQRS ? (1) x = 30 degree  [#permalink]

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27 Aug 2015, 00:37
3
VenoMfTw wrote:
VeritasPrepKarishma

If in a quadrilateral if one pair of opposite angles are equal then can we conclude that it will be a parallelogram?
In this example Y = 120 and QRS = 120. Can we conclude that QRST is a parallelogram?
Attachment:
The attachment SASASA.png is no longer available

No, you cannot. Look at this diagram:

Attachment:

Geometrical Figures.png [ 44.31 KiB | Viewed 1708 times ]

from
https://commons.wikimedia.org/wiki/File ... square.svg

Note the Kite has east and west angles equal but it is not a parallelogram.
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Re: What is the perimeter of PQRS ? (1) x = 30 degree  [#permalink]

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06 Feb 2016, 18:28
Forget conventional ways of solving math questions. In DS, Variable approach is the easiest and quickest way to find the answer without actually solving the problem. Remember equal number of variables and independent equations ensures a solution.

What is the perimeter of PQRS ?
(1) x = 30 degree
(2) w= 45 degree

When you modify the condition and the question, you can figure out PQ is an angle of 60 degrees and the length of PQ from QT=2, PT=1. Also, since y=120, you can come up with QS. Thus, you only need to figure out RS, which makes 1 variable. You also need 1 equation in order to match with the number of equations. For 1) 1 equation, for 2) 1 equation, which is likely to make D the answer.
For 1), you already know that x=30, which is not sufficient.
For 2), from w=45, you can figure out the perimeter, which is sufficient. Therefore, the answer is B.

-> For cases where we need 1 more equation, such as original conditions with “1 variable”, or “2 variables and 1 equation”, or “3 variables and 2 equations”, we have 1 equation each in both 1) and 2). Therefore, there is 59 % chance that D is the answer, while A or B has 38% chance and C or E has 3% chance. Since D is most likely to be the answer using 1) and 2) separately according to DS definition. Obviously there may be cases where the answer is A, B, C or E.
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Re: What is the perimeter of PQRS ? (1) x = 30 degree  [#permalink]

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09 Feb 2016, 11:59
VeritasPrepKarishma wrote:
hi everyone ,

this was a good question and very well explained by blueseas. I had only one question as blueseas himself states
Quote:
One of the possibility is P=90 X=30 Hence in that case triangle PQT is right triangle with angle 90:60:30...hence side will be in ratio 2:\sqrt{3}:1

so this is just one case. Is it good enough to answer it as B. I mean what if P+X = 89+ 31,, then shouldnt the perimeter be different?

A triangle is uniquely defined by two sides and the included angle. Given two sides and the included angle, the third side and the other two angles are fixed. So there is only one possibility.

And we assume it is a right triangle based on the given sides of 1 and 2, the angle 60, and the two other angles combined are 120.
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Re: What is the perimeter of PQRS ? (1) x = 30 degree  [#permalink]

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15 Mar 2016, 07:26
consider QRST, quadrilateral, by keeping angle R=120 and changing angle RQT and RST, we can draw different possibilities ( that means we can move point R here and there), whereas in triangle PQT, one side PT=1 is fixed and one angle QTP is fixed, so we cant move point P here and there.

So to find perimeter, we need to think of fixing the boundaries
since PQT triangle is already fixed, knowing X degrees won't help
on the other hand in order to fix point R , a specific value of W, will fix QRST
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Re: What is the perimeter of PQRS ? (1) x = 30 degree  [#permalink]

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21 Mar 2016, 18:39
Can anybody explain in detail the calculation of QR and RS and the actual values of both? I understand that we need not calculate the values on actual GMAT but for just to know the how part of calculation I request someone to post the calculation for QR and RS. Remaining part is fine for me.
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Re: What is the perimeter of PQRS ? (1) x = 30 degree  [#permalink]

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26 Jul 2016, 02:43
I understand now that under statement two, we can prove that all the lines are fixed. But how would we go about finding the actual lengths of QR and RS?

Do we just accept it as a rule that if you know all of the angles of a quadrilateral and the length of two of its sides, you could calculate the lengths of its other two sides?
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Re: What is the perimeter of PQRS ? (1) x = 30 degree  [#permalink]

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26 Jul 2016, 23:00
1
momonmoprob wrote:
I understand now that under statement two, we can prove that all the lines are fixed. But how would we go about finding the actual lengths of QR and RS?

Do we just accept it as a rule that if you know all of the angles of a quadrilateral and the length of two of its sides, you could calculate the lengths of its other two sides?

In quadrilateral TQRS, angle Y = angle R = 120 degrees
angle W = 45 degrees

Sum of all 4 angles of a quad = 360 = 120 + 120 + 45 + angle TQR

Angle TQR = 45 degrees.

So the angles are 45, 45, 120, 120. The opposite angles are equal so this is a parallelogram.

So QR is parallel and equal to TS.
TS = 3 = QR

Also, QT is parallel and equal to RS
QT = 2 = RS
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Re: What is the perimeter of PQRS ? (1) x = 30 degree  [#permalink]

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27 Jul 2016, 08:07
VeritasPrepKarishma wrote:
momonmoprob wrote:
I understand now that under statement two, we can prove that all the lines are fixed. But how would we go about finding the actual lengths of QR and RS?

Do we just accept it as a rule that if you know all of the angles of a quadrilateral and the length of two of its sides, you could calculate the lengths of its other two sides?

In quadrilateral TQRS, angle Y = angle R = 120 degrees
angle W = 45 degrees

Sum of all 4 angles of a quad = 360 = 120 + 120 + 45 + angle TQR

Angle TQR = 45 degrees.

So the angles are 45, 45, 120, 120. The opposite angles are equal so this is a parallelogram.

So QR is parallel and equal to TS.
TS = 3 = QR

Also, QT is parallel and equal to RS
QT = 2 = RS

360 = 120 + 120 + 45 + TQR
TQR = 360 - 120 - 120 - 45
TQR = 75

Am I missing something?
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Re: What is the perimeter of PQRS ? (1) x = 30 degree  [#permalink]

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28 Jul 2016, 04:55
How did you get PQ using these formulas only using only w = 45 ?

2/sin(P) = 1/sin(X) = PQ/sin(60)

2/1 = sin(60)/sin(X) -----> get X, hence get PQ ?
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Re: What is the perimeter of PQRS ? (1) x = 30 degree  [#permalink]

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16 Jan 2017, 06:43
blueseas wrote:
Asifpirlo wrote:
What is the perimeter of PQRS ?
(1) x = 30 degree
(2) w= 45 degree

need some alternative ways

IN DS we dont need to calculate the exact value only we should make sure that answer is definite.

now in this question perimeter of PQRS is asked.
since this is quadrilateral hence sum of interior angles will be 360

statement 1:
$$x= 30$$
triangle QPT right triangle with angles 90 60 30...hence we can find out PQ
ANGLE Y = 120
Now if you see quad QRST ==>angle W and angle (Q-X) are not fixed...so to make angle R=120 we can make different figures. and hence there will be different perimeters.
hence insufficient.

statement 2:
w=45 now as quad QRST ==>sum of interior angle = 360
hence angle Q-X= 75
now since we have 2 sides fixed(QT and ST) and four angles are fixed...hence we are going to get a unique quad.hence we can calculate the length of QR and RS.

Now we have to find out only the length of PQ to determine the perimeter of quad PQRS.
SUM of interior angles of quad PQRS will be 360 hence ...P + W +120+Q =360
Now since Q= 75+X AND W=45
Putting the values and solving ...we get P+X = 120

NOW since P+X =120
One of the possibility is P=90 X=30 Hence in that case triangle PQT is right triangle with angle 90:60:30...hence side will be in ratio $$2:\sqrt{3}:1$$
now this clearly satisfies that triangle PQT is 90/60/30 TRIANGLE hence SIDE PQ = $$\sqrt{3}$$

HENCE we can find out the perimeter.hence sufficient

Hello, can you elaborate in st.2 how do we take for granted that Q=45 degrees?
I mean we do not know that Q=W or that R=Y. It is not stated that PS is a straight line, how can we assume that Y=180-60=120?
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Re: What is the perimeter of PQRS ? (1) x = 30 degree  [#permalink]

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16 Jan 2017, 20:40
1
arven wrote:
blueseas wrote:
Asifpirlo wrote:
What is the perimeter of PQRS ?
(1) x = 30 degree
(2) w= 45 degree

need some alternative ways

IN DS we dont need to calculate the exact value only we should make sure that answer is definite.

now in this question perimeter of PQRS is asked.
since this is quadrilateral hence sum of interior angles will be 360

statement 1:
$$x= 30$$
triangle QPT right triangle with angles 90 60 30...hence we can find out PQ
ANGLE Y = 120
Now if you see quad QRST ==>angle W and angle (Q-X) are not fixed...so to make angle R=120 we can make different figures. and hence there will be different perimeters.
hence insufficient.

statement 2:
w=45 now as quad QRST ==>sum of interior angle = 360
hence angle Q-X= 75
now since we have 2 sides fixed(QT and ST) and four angles are fixed...hence we are going to get a unique quad.hence we can calculate the length of QR and RS.

Now we have to find out only the length of PQ to determine the perimeter of quad PQRS.
SUM of interior angles of quad PQRS will be 360 hence ...P + W +120+Q =360
Now since Q= 75+X AND W=45
Putting the values and solving ...we get P+X = 120

NOW since P+X =120
One of the possibility is P=90 X=30 Hence in that case triangle PQT is right triangle with angle 90:60:30...hence side will be in ratio $$2:\sqrt{3}:1$$
now this clearly satisfies that triangle PQT is 90/60/30 TRIANGLE hence SIDE PQ = $$\sqrt{3}$$

HENCE we can find out the perimeter.hence sufficient

Hello, can you elaborate in st.2 how do we take for granted that Q=45 degrees?
I mean we do not know that Q=W or that R=Y. It is not stated that PS is a straight line, how can we assume that Y=180-60=120?

Lines that look straight in diagrams are straight.
Anyway, you are asked the perimeter of PQRS so PS must be a straight line. Else, the question would have asked for the perimeter of the figure PQRST.
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Re: What is the perimeter of PQRS ? (1) x = 30 degree &nbs [#permalink] 16 Jan 2017, 20:40

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