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

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

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30 Jan 2017, 10:21
Can some one explain why in in Statement II - how can we determine length of QR and RS so that the perimeter is fixed?..

I know we do not need a nomber value answer is DS. But I am not able to understand the visualisations in the above thread.
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Re: What is the perimeter of PQRS ? (1) x = 30 degree  [#permalink]

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28 Mar 2017, 22:35
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

I still don't understand how are you concluding that QR =TS and QT= RS

Aren't angles supposed to be 75-45-120-120 to equal 360 in QRST ?

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

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05 May 2017, 03:09
1
hello folks
let me give my 2 cents why B is sufficient
in quad QRST, w = 45 as sum of interior angle of a quad is 360 so we come to know the value of angle Q=75. hence we know all sides
in angle QPT , we know 2 sides TQ and TP and angle between these sides . hence its is easy to calculate side opposite to this.
now you know all sides. Sum all and get perimeter.
Hence B is right
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Re: What is the perimeter of PQRS ? (1) x = 30 degree  [#permalink]

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28 Jun 2017, 11:01
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

The other possibility is of the Angle P and X to be 60* each, not giving us an exact value and hence should be insufficient, making us look for other options.
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What is the perimeter of PQRS ? (1) x = 30 degree  [#permalink]

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28 Jun 2017, 11:18
Guys blueseas, Asifpirlo, VeritasPrepKarishma,

***** I disagree with the explanations given above as a solution. Maybe I am wrong, so please point out my mistakes if any. *****

Let me start from the given information. Please consider * as a sign for Degree.

Known Info: Angle QTP = 60*
Angle R = 120*
Side TS = 3
Side QT = 2
Side PT = 1

We Need: Side QR, Side RS, {Value of Angle QTS, Value of Angle P (For Calculations, if any)}

Now, Let's get started.

Statement 1: x = 30* {AD be the possible solutions.}

Now if everyone is aware of the Triangle theory for 30* - 60* - 90* Triangle, then it states that the value of side subtending the Angles 90* and 60* have the value, let's say "a" and the subsequent side subtending Angle 90* and 30* has value "a.sq.root3" and Hypotenuse has a value of "2a". Attachment below.

So we have "a" as 1 in this case, representing side PT = 1(from given info).

Now the One of the angle subtended on this side is 60*. Angle PQT (Angle x) is given to be 30*. The Angle P becomes 90*( Total sum of all the angles in a Triangle is 180*).

So PT = a = 1.

QT = 2a = 2 (Already given in question info).

Hence, QP = Sq.root/3 (As per the theory explained above). We got one of the Unknown side. Proceeding to find remaining required information.

*****Now coming to the other side of the diagram.*****

We had Angle QTP = 60* (Given info)
So Angle QTS = 120* ( Line bisected by other line divides the 180* angle.)

*** So, Angle QTS = Angle R.

From the line theory, if the opposite angles subtended by intersecting lines are equal then they are said to be parallel. Which means that QT/RS are parallel and QR/TS as well as parallel for them to subtend equal opposite angles of 120*.

Since they are parallel, then the unknown Angles of W and TQR should even be equal, making the four sided polygon to be called as PARALLELOGRAM, with no angle as 90*( for it to be called a Rectangle).

So, QR becomes 3 and we RS becomes 2, giving us the required information. Hence A is sufficient. Looking for if D could be other correct option if the information in Statement 2 becomes equally sufficient.
-------------------------------------------
Coming to Statement 2:

W = 45*

Now Angle R is already = 120* and W is given to be = 45*, which on initial inspection does not satisfy the line theory of internal angles subtended by intersecting lines to be 180*.

Ok for an instance, let's consider the figure is not drawn to scale.
So we require the value of Angle QTS, which is known to be = 120* (As explained before).

Total Sum of all the angles in a 4 sided polygon is = (n-2)180*, where n = 4, is equal to 360*.

We have value of three angles and now to calculate the value of Angle TQR;

120* + 45* + 120* + TQR = 360*.
Solving, TQR = 75*.

This makes it highly unlikely to find out exact values of Angle PQT and Angle P, and finally, the side PQ cannot be determined.

Hence, D goes out and A is the Answer.
Attachments

Angle Theory.jpg [ 23.71 KiB | Viewed 858 times ]

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

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29 Jun 2017, 01:53
2
msonnie17 wrote:
Guys blueseas, Asifpirlo, VeritasPrepKarishma,

***** I disagree with the explanations given above as a solution. Maybe I am wrong, so please point out my mistakes if any. *****

Let me start from the given information. Please consider * as a sign for Degree.

Known Info: Angle QTP = 60*
Angle R = 120*
Side TS = 3
Side QT = 2
Side PT = 1

We Need: Side QR, Side RS, {Value of Angle QTS, Value of Angle P (For Calculations, if any)}

Now, Let's get started.

Statement 1: x = 30* {AD be the possible solutions.}

Now if everyone is aware of the Triangle theory for 30* - 60* - 90* Triangle, then it states that the value of side subtending the Angles 90* and 60* have the value, let's say "a" and the subsequent side subtending Angle 90* and 30* has value "a.sq.root3" and Hypotenuse has a value of "2a". Attachment below.

So we have "a" as 1 in this case, representing side PT = 1(from given info).

Now the One of the angle subtended on this side is 60*. Angle PQT (Angle x) is given to be 30*. The Angle P becomes 90*( Total sum of all the angles in a Triangle is 180*).

So PT = a = 1.

QT = 2a = 2 (Already given in question info).

Hence, QP = Sq.root/3 (As per the theory explained above). We got one of the Unknown side. Proceeding to find remaining required information.

*****Now coming to the other side of the diagram.*****

We had Angle QTP = 60* (Given info)
So Angle QTS = 120* ( Line bisected by other line divides the 180* angle.)

*** So, Angle QTS = Angle R.

From the line theory, if the opposite angles subtended by intersecting lines are equal then they are said to be parallel. Which means that QT/RS are parallel and QR/TS as well as parallel for them to subtend equal opposite angles of 120*.

-------------------------------------------

Not correct. Think of a kite.
Angle A = angle C but that doesn't mean opposite sides are parallel. Stmnt 1 is not sufficient alone.

Attachment:

Kite.gif [ 2.92 KiB | Viewed 853 times ]

Quote:
Coming to Statement 2:

W = 45*

Now Angle R is already = 120* and W is given to be = 45*, which on initial inspection does not satisfy the line theory of internal angles subtended by intersecting lines to be 180*.

Ok for an instance, let's consider the figure is not drawn to scale.
So we require the value of Angle QTS, which is known to be = 120* (As explained before).

Total Sum of all the angles in a 4 sided polygon is = (n-2)180*, where n = 4, is equal to 360*.

We have value of three angles and now to calculate the value of Angle TQR;

120* + 45* + 120* + TQR = 360*.
Solving, TQR = 75*.

This makes it highly unlikely to find out exact values of Angle PQT and Angle P, and finally, the side PQ cannot be determined.

Hence, D goes out and A is the Answer.

You cannot leave at "highly unlikely". Check out my previous solution to see why stmtnt 2 is sufficient.
https://gmatclub.com/forum/what-is-the- ... l#p1294806
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Re: What is the perimeter of PQRS ? (1) x = 30 degree  [#permalink]

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23 Nov 2017, 03:38
1
VeritasPrepKarishma wrote:
Asifpirlo wrote:
What is the perimeter of PQRS ?
(1) x = 30 degree
(2) w= 45 degree

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

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

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

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.

Responding to a pm:

Quote:
In a typical DS question, we are expected to know the values of the unknowns to decide whether a statement is sufficient. In this case, all we know is that the other sides (PQ, QR, RS and SP) have fixed wrt to each other. How does that help to get the lengths?

We found that "PQ, QR, RS and SP all have a defined length." So their lengths are fixed and can be found out. So if we try to find their lengths, we will get that PQ is say 1.5, QR is say 5 (just random values) etc. So the data is sufficient to find the perimeter. We don't really have to find the actual perimeter since it is a sufficiency question. Just saying whether we can find or not is sufficient.
In a DS question, we don't need to know the value of an unknown. We need to know that a unique value exists. What the actual value is doesn't matter.
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Re: What is the perimeter of PQRS ? (1) x = 30 degree  [#permalink]

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26 Nov 2017, 12:09
1
Important points:
1. For DS questions we don't have to find exact value. We just need to check if we can find 1 unique answer
2. In geometry , don't ever assume 2 lines to be parallel until given and don't assume figure to be of a definite shape. In this question, for given figure QRTS might look like a parallelogram but don't assume it to be a parallelogram. ALso angle QTS can be calculated = 120 = angle QST. SO don''t assume angle TQR to be equal to angle TSR.

Now, we have 2 very important trigonometry formulas:

1. $$Cos C = \frac{a^2+b^2-c^2}{2ab}$$
2. $$\frac{a}{Sin A} = \frac{b}{Sin B} = \frac{c}{Sin C}$$

where a,b and c are sides of the triangle and A,B and C are angles of triangle opposite to sides a,b and c respectively. That is , there are 6 variables

Do we have to remember the formula. Not as such. Using the above 2 formulas we can calculate all angels and all sides if we know 3 variables. Thus we can say :

Rule: If we know any 3 variables out of these 6 variables we can calculate all the remaining variables. Only exception is that all the known variables cannot be all angles. We need measure of at least 1 side

(You can try to verify the rule )

So now lets come back to the question:

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

What we have to find : PQ+QR+RS+TS+PT. we already know value of PT and TS from figure. So we have to find remaining sides: PQ+QR+RS
Before moving to given statements lets try to find out what can we conclude from given figure:

1. Triangle PQT : we know 1 angle and 2 sides => we can calculate all sides and all angles of the triangle. So now we know measure of PQ, angle P and angle PQT
2. Triangle QTS : we can calculate angle QTS = 120. So now we know 2 sides and 1 angle of the triangle , thus we can calculate remaining variables. So know the value of QS, angle TQS and angle TSQ
3. Triangle QRS : we know 1 angle QRS and 1 side QS. here we need at least 1 more variable to find the remaining variables.

So now lets move to given data:

Statement 1 : x = 30 degree.
As we can calculate angle x using the above method, this statement doesn't give us any additional information. Also we need information in triangle QRS to calculate QR and RS.
Not suffiicent

Statement 2 : w = 45 degree.
As w = 45 degree, we can calculate angle QSR.
Angle QSR = W - angle QST
So now for triangle QRS we know value of 3 variables, QS, angle QRS and angle QSR, thus we can calculate QR, RS and angle SQR.

Hence we got the value of all the sides of the given figure.
Sufficient

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

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04 Feb 2018, 22:54
M not getting solutions. Can some one please explain in detail

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

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04 Feb 2018, 22:57
ketkichaudhary1992 wrote:
M not getting solutions. Can some one please explain in detail

Sent from my [device_name] using [url]GMAT Club Forum mobile app[/url]

Please check previous THREE pages of the discussion. If something remains unclear, please ask specific questions. Thank yuo.
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Re: What is the perimeter of PQRS ? (1) x = 30 degree  [#permalink]

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24 Mar 2018, 09:38
mau5 wrote:
Asifpirlo wrote:
What is the perimeter of PQRS ?
(1) x = 30 degree
(2) w= 45 degree

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
Attachment:
2.png

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

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

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.

This is an excellent way of visualization, made things clearer. However, the first scenario as part of statement 1 does not represent 120 degrees does it? I know the bigger point is that W is variable but it's hard to see that when the PQ. PT and TS are fixed and you're still trying to imagine an obtuse angle around R. Does that make sense or is there a better of way of thinking about this?
Re: What is the perimeter of PQRS ? (1) x = 30 degree &nbs [#permalink] 24 Mar 2018, 09:38

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