An alternative way is to actually treat it as a DS question. Take the given info and try to find out whether you will have a fixed perimeter of PQRS. Try to draw PQRS from the given info and find out whether the perimeter has a unique value.

Let me show you why statement 2 alone is sufficient.
Given data:
Angle QRS = 120 degrees
ST = 3
PT = 1
QT = 2
Angle QTP = 60 degrees
Angle RST = w = 45 degrees (from statement 2)

Let's start with angle QRS. Make an angle of 120 degrees. We don't know the lengths of QR and RS so we don't know where Q and S will lie. Now we have to make the line ST such that the angle is 45 degrees. Many such lines are possible since we don't know where S lies. We have made 3 such lines as an example.


Now, T will lie 3 units down one of these lines. Angle QTP is 60 degrees so you make a 60 degree line.



But we also know that QT is 2 so QT will be 2 in only one of these cases i.e. for only one of these lines we made as ST, QT will be 2. So you will get a unique point for S, T and Q. Now 1 unit down the line of ST, make P. Join it to Q.

So PQ, QR, RS and SP all have a defined length. Hence the perimeter of PQRS can be found given this data. Hence, statement (2) alone is sufficient.

Similarly, try to figure out why statement 1 alone is not sufficient.
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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

hope i have made it clear...ask if something is not clear.

yaa got it.. i have a solution too which is similar too.. but i used trigonometry for this problem.. will upload the solution...... but you did a great job once again... nice.....

blue seas,

how do we calculate QR & RS if we know all the angles of a quadrilateral?

This is my solution friends.....

I always use this two formulas for hard problems.
1. cosC = a^+b^-c^2/2ab
2. a/sinA = b/sinB = c/sinC

Now from statement (1), we know x = 30 .
but from the triangle PQT we can easily derive this value of PQ using formula 1.
Then by using formula (2) into that triangle we can evaluate the value of x .

So statement (1) is one that we already know and thus insufficient.

So now we know the value of PQ and PS .
We need just QR and RS .

From triangle QTS we can evaluate QS by using formula (1) .
Y=120 degrees.
After that from triangle QTS we can evaluate angle QST by using formula (2)

From statement(2) we know W=45 degree. So subtracting angles QST from W will give us the value of angle QSR.

Now in triangle QRS we have two known angles and one side. By applying formula (2) we can evaluate the rest of the two sides and finally evaluate the perimeter….

So statement (2) alone is sufficient. Answer is (B) .

{we don’t need to evaluate the values here because its ds problems. Using the two formula I mentioned can provide us just yes or no and thus we can evaluate the answer easily and fast, hope this works}

at the top of the post beside timer.......
got it man? if not then tell me..no problem... i will help you to find... welcome to gmatclub

i see the options to post the answers however I don't see the options TO the answers. I see A B C D E as possible response answers but what exactly I am answering I don't know.

I don't see A. xyzz answer B. this C. that and then the option of whether A, B or C is correct

A B C D E all have a fixed meaning in data sufficiency brother,thats why its not important to describe A B C D E in every questions,You can have the meaning behind these letters in this websites too. just search the basic Data sufficiency topics there You will have it all.

hi michael,

this is a data sufficiency question.
please refer to below link to undersatand the rules for solving these types of question.
http://www.gmathacks.com/data-sufficien ... -test.html

hope this helps

hey blueseas, I like your solution.. we are asked to find the perimeter of pqrs... so we found pq=sqrt(3) however what about the QR and RS? what are their lengths?

So please elaborate on the "one of the possibilities" condition. Are there other real possibilities? In those other possibilities, how do we know the perimeter will not be different? Which would yield answer E.

I think that all we manage to prove here that the shape is a quadrilateral (as in not a parallelogram). thus we cannot assume that opposite sides are equal.. in which case the answer would be E?

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

This question is a perfect example to try and visualise the question.

Worth Remembering :

You always need two different lines to define an angle.

From F.S 1, we try creating the traingle PQT. You have two line segments, PT and QT of fixed lengths.Now, when they are placed end to end at point T,with an angle of 60 degrees between them, there will be ONLY ONE line segment of unique length between P and Q, i.e. line segment PQ.Thus, we know the length of PQ.Thus, this fact statement adds nothing new to find the length of PQ.

Now, imagine constructing the other part of the figure, the quadrilateral QRST. We place the line segment TS of length 3 units, with an angle of (180-60 = 120 degrees) between TS and QT.

Refer to the 2 images. We can see that depending upon 2 values of w, we will have 2 different lenghts of QR and RS,which will esentially change our perimeters. Thus, this fact statement is Insufficient.

Again, from F.S 2, we know that w=45 degrees.
Now, the let the line SC, at an angle of 45 degrees, extend till infinity.

Now, to intersect the line segment SC at 120 degrees, lets construct another line CD, which also extends till infinity.

Now, imagine sliding the entire line CD,downwards, and there will be a single point when it will meet the point Q. Thus, we would have fixed lengths for QR,RS and PQ we had already established needed no additional details to be calculated.Sufficient.

Might be a bit daunting at first, but it might help once you realise this method of mind-visualization.
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Given A,
when we have that both R and Y are 120, does that not give us parallelism between TS and QR
and between RS and TQ?

NO they would be parallel only if w=60 degrees.
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hi everyone ,

this was a good question and very well explained by blueseas. I had only one question as blueseas himself states


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?

P=90 and X=30 is not ONE OF a possibility.It is the ONLY possibility.There are no other values for P and X that will satisfy the given triangle PQT.
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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.
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