What is the perimeter of PQRS ? (1) x = 30 degree : GMAT Data Sufficiency (DS)
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# What is the perimeter of PQRS ? (1) x = 30 degree

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

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05 Aug 2013, 02:15
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What is the perimeter of PQRS ?
(1) x = 30 degree
(2) w= 45 degree

[Reveal] Spoiler:
need some alternative ways
[Reveal] Spoiler: OA

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

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05 Aug 2013, 03:10
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Asifpirlo wrote:
What is the perimeter of PQRS ?
(1) x = 30 degree
(2) w= 45 degree

[Reveal] Spoiler:
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

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

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05 Aug 2013, 04:37
2
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blueseas wrote:
Asifpirlo wrote:
What is the perimeter of PQRS ?
(1) x = 30 degree
(2) w= 45 degree

[Reveal] Spoiler:
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

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

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05 Aug 2013, 05:12
blueseas wrote:
Asifpirlo wrote:
What is the perimeter of PQRS ?
(1) x = 30 degree
(2) w= 45 degree

[Reveal] Spoiler:
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

blue seas,

how do we calculate QR & RS if we know all the angles of a quadrilateral?
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Re: What is the perimeter of PQRS ? (1) x = 30 degree [#permalink]

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05 Aug 2013, 05:52
3
KUDOS
blueseas wrote:
Asifpirlo wrote:
What is the perimeter of PQRS ?
(1) x = 30 degree
(2) w= 45 degree

[Reveal] Spoiler:
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

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

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05 Aug 2013, 08:42
drmichael wrote:
Asifpirlo wrote:
What is the perimeter of PQRS ?
(1) x = 30 degree
(2) w= 45 degree

I don't see the options A , B , C, D etc. How come the timer asks for options but all I see are (1) and (2) where are the answer options?
[Reveal] Spoiler:
need some alternative ways

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

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05 Aug 2013, 09:25
Asifpirlo wrote:
drmichael wrote:
Asifpirlo wrote:
What is the perimeter of PQRS ?
(1) x = 30 degree
(2) w= 45 degree

I don't see the options A , B , C, D etc. How come the timer asks for options but all I see are (1) and (2) where are the answer options?
[Reveal] Spoiler:
need some alternative ways

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

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05 Aug 2013, 10:29
Asifpirlo wrote:
What is the perimeter of PQRS ?
(1) x = 30 degree
(2) w= 45 degree

[Reveal] Spoiler:
need some alternative ways

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

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05 Aug 2013, 10:34
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drmichael wrote:
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

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

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19 Nov 2013, 10:50
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?
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Re: What is the perimeter of PQRS ? (1) x = 30 degree [#permalink]

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19 Nov 2013, 20:35
blueseas wrote:
Asifpirlo wrote:
What is the perimeter of PQRS ?
(1) x = 30 degree
(2) w= 45 degree

[Reveal] Spoiler:
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

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

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19 Nov 2013, 23:15
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?
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Re: What is the perimeter of PQRS ? (1) x = 30 degree [#permalink]

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19 Nov 2013, 23:17
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Asifpirlo wrote:
What is the perimeter of PQRS ?
(1) x = 30 degree
(2) w= 45 degree

[Reveal] Spoiler:
need some alternative ways

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.

Attachment:

Ques4.jpg [ 11.29 KiB | Viewed 14432 times ]

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

Attachment:

Ques5.jpg [ 14.34 KiB | Viewed 14430 times ]

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.
Attachment:

Ques6.jpg [ 9.98 KiB | Viewed 14414 times ]

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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Get started with Veritas Prep GMAT On Demand for $199 Veritas Prep Reviews Senior Manager Joined: 07 Apr 2012 Posts: 464 Followers: 2 Kudos [?]: 53 [1] , given: 58 Re: What is the perimeter of PQRS ? (1) x = 30 degree [#permalink] ### Show Tags 21 Nov 2013, 02:20 1 This post received KUDOS blueseas wrote: Asifpirlo wrote: What is the perimeter of PQRS ? (1) x = 30 degree (2) w= 45 degree [Reveal] Spoiler: 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 hope i have made it clear...ask if something is not clear. How do we know this? "hence angle Q-X= 75" Verbal Forum Moderator Joined: 10 Oct 2012 Posts: 630 Followers: 80 Kudos [?]: 1116 [3] , given: 136 Re: What is the perimeter of PQRS ? (1) x = 30 degree [#permalink] ### Show Tags 21 Nov 2013, 09:54 3 This post received KUDOS 4 This post was BOOKMARKED Asifpirlo wrote: What is the perimeter of PQRS ? (1) x = 30 degree (2) w= 45 degree [Reveal] Spoiler: need some alternative ways 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. Attachment: SASASA.png [ 8.17 KiB | Viewed 14348 times ] Attachment: 2.png [ 8.25 KiB | Viewed 14336 times ] 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. Attachment: 3.png [ 8.72 KiB | Viewed 14341 times ] Now, to intersect the line segment SC at 120 degrees, lets construct another line CD, which also extends till infinity. Attachment: 4.png [ 10.26 KiB | Viewed 14336 times ] 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. _________________ Senior Manager Joined: 07 Apr 2012 Posts: 464 Followers: 2 Kudos [?]: 53 [0], given: 58 Re: What is the perimeter of PQRS ? (1) x = 30 degree [#permalink] ### Show Tags 23 Nov 2013, 02:42 Asifpirlo wrote: What is the perimeter of PQRS ? (1) x = 30 degree (2) w= 45 degree [Reveal] Spoiler: need some alternative ways 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? Verbal Forum Moderator Joined: 10 Oct 2012 Posts: 630 Followers: 80 Kudos [?]: 1116 [0], given: 136 Re: What is the perimeter of PQRS ? (1) x = 30 degree [#permalink] ### Show Tags 23 Nov 2013, 03:07 ronr34 wrote: Asifpirlo wrote: What is the perimeter of PQRS ? (1) x = 30 degree (2) w= 45 degree [Reveal] Spoiler: need some alternative ways 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. _________________ Intern Joined: 07 Jan 2013 Posts: 43 Location: India Concentration: Finance, Strategy GMAT 1: 570 Q46 V23 GMAT 2: 710 Q49 V38 GPA: 2.9 WE: Information Technology (Computer Software) Followers: 1 Kudos [?]: 23 [0], given: 23 Re: What is the perimeter of PQRS ? (1) x = 30 degree [#permalink] ### Show Tags 24 Nov 2013, 04:30 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? _________________ Help with Kudos if I add to your knowledge realm. Verbal Forum Moderator Joined: 10 Oct 2012 Posts: 630 Followers: 80 Kudos [?]: 1116 [0], given: 136 Re: What is the perimeter of PQRS ? (1) x = 30 degree [#permalink] ### Show Tags 24 Nov 2013, 04:37 adg142000 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? 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. _________________ Veritas Prep GMAT Instructor Joined: 16 Oct 2010 Posts: 7118 Location: Pune, India Followers: 2129 Kudos [?]: 13625 [1] , given: 222 Re: What is the perimeter of PQRS ? (1) x = 30 degree [#permalink] ### Show Tags 24 Nov 2013, 19:29 1 This post received KUDOS Expert's post 1 This post was BOOKMARKED adg142000 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. _________________ Karishma Veritas Prep | GMAT Instructor My Blog Get started with Veritas Prep GMAT On Demand for$199

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Re: What is the perimeter of PQRS ? (1) x = 30 degree   [#permalink] 24 Nov 2013, 19:29

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