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805+ Level|   Geometry|      
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What is the perimeter of PQRS ? (1) x = 30 degree (2) w= 45 degree 28 Jun 2017, 12:18
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Guys blueseas, Asifpirlo, VeritasPrepKarishma,

cabgshriram: This explanation might help you too.


***** 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
Angle Theory.jpg [ 23.71 KiB | Viewed 4865 times ]

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What is the perimeter of PQRS ? (1) x = 30 degree (2) w= 45 degree 29 Jun 2017, 02:53
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Guys blueseas, Asifpirlo, VeritasPrepKarishma,

cabgshriram: This explanation might help you too.


***** 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*.


-------------------------------------------
Show more


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
Kite.gif [ 2.92 KiB | Viewed 4900 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.
Show more

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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What is the perimeter of PQRS ? (1) x = 30 degree (2) w= 45 degree 23 Nov 2017, 04:38
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What is the perimeter of PQRS ?
(1) x = 30 degree
(2) w= 45 degree

Show 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
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.
Show more

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?
Show more

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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What is the perimeter of PQRS ? (1) x = 30 degree (2) w= 45 degree 26 Nov 2017, 13:09
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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 :P )

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

Answer: B
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What is the perimeter of PQRS ? (1) x = 30 degree (2) w= 45 degree 16 Feb 2022, 11:27
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Asifpirlo

What is the perimeter of PQRS ?

(1) x = 30 degree
(2) w= 45 degree

Show Spoiler
Attachment:
The attachment SASASA.png is no longer available
Attachment:
The attachment 2019-07-19_1032.png is no longer available
Show more

Statement 1:
Since x is 30, triangle PQT becomes a 30-60-90 triangle, hence PQ = √3
However, we cannot find the lengths of QR or RS since QTSR is not necessarily a parallelogram
Note: though the opposite angles R and angle QTS are 120 each, its not enough to conclude its a parallelogram
This is because there can be multiple positions of R where the value of the angle will be 120. In these cases, the lengths of R and RS will be different (the diagram below explains why):

Attachment:
WhatsApp Image 2022-02-16 at 11.37.21 PM.jpeg
WhatsApp Image 2022-02-16 at 11.37.21 PM.jpeg [ 99.5 KiB | Viewed 2058 times ]


Thus, it is not sufficient


Statement 2:

Since we know that angle RSP is 45, we can drop the perpendicular from Q on PS as shown in the diagram below:

Attachment:
WhatsApp Image 2022-02-16 at 11.58.49 PM.jpeg
WhatsApp Image 2022-02-16 at 11.58.49 PM.jpeg [ 97.3 KiB | Viewed 2055 times ]

Triangle QXT is 30-60-90 => We can determine QX (= √3) and TX (= 1) implying that P must overlap with X
Thus, PQ = √3

In QTSR: Three angles are: R = 120, S = 45 (given), angle QTS = 180-60 = 120
Thus, the 4th angle is angle RQT = 75
Also, the sides QT and TS are given. So, the quadrilateral is defined and unique.

Why: Start by drawing QT = 2. Make an angle of 120 at T and draw TS = 3. At S, draw angle TSN = 45 and at Q, draw angle MQT = 75
These lines QM and SN will now meet at R => We have defined quadrilateral QTSR
Thus, the lengths of QR and RS are defined
=> Perimeter of PQRS is defined

Thus, statement 2 is sufficient

Answer B
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What is the perimeter of PQRS ? (1) x = 30 degree (2) w= 45 degree 18 Oct 2022, 05:40
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I am very confused here, most of the solutions are answering the wrong quesitons i.e. can you find the perimeter of the shape PQRS. that is not the question the question is what is the perimeter i.e. given the info provided can you find the answer, in this case can you find the legnths of PQ, QR, RS. With out using trigronmentry you can not find QR or RS ( PQ can be solved for using triangle properties) so E should be OA.

Bunuel please help
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What is the perimeter of PQRS ? (1) x = 30 degree (2) w= 45 degree 19 Oct 2022, 10:06
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Tshenolo
I am very confused here, most of the solutions are answering the wrong quesitons i.e. can you find the perimeter of the shape PQRS. that is not the question the question is what is the perimeter i.e. given the info provided can you find the answer, in this case can you find the legnths of PQ, QR, RS. With out using trigronmentry you can not find QR or RS ( PQ can be solved for using triangle properties) so E should be OA.

Bunuel please help
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From Official Guide:

The data sufficiency problem consists of a question and two statements, labeled (1) and (2), in which certain data are given. You have to decide whether the data given in the statements are sufficient for answering the question. ...

So, for a data sufficiency problem, you should decide whether the data given in the statements are enough for answering the question, you don't have to actually solve and get the answer...

For example, consider this DS, I just made up:

If n is a positive integer, what is the \(n^{th}\) prime number ?

(1) \(n = 1,000,001\)
(2) \((n - 1)^{th}\) prime number is 1,5485,863

The answer to the above question is D: EACH statement ALONE is sufficient to answer the question asked. Because knowing the value of n, it's possible to get the answer and knowing that 15485863 is \((n - 1)^{th}\) prime, it's possible to get the next prime. But I don't think anyone can possibly know or calculate during the exam that 1,000,001st prime is 1,5485,867 or that the next prime after 1,5485,863 is actually 1,5485,867.

Hope it helps.
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What is the perimeter of PQRS ? (1) x = 30 degree (2) w= 45 degree 07 Feb 2023, 08:04
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KarishmaB
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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
Show more

Hey Karishma B, as per calculations(360-240-45=75) angle TQR will be 75, not 45, hence quad TQRS is not parallelogram, then how can we determine sides QR and RS ?
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What is the perimeter of PQRS ? (1) x = 30 degree (2) w= 45 degree 08 Jan 2024, 21:37
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KarishmaB
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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
Show more



Hi, angle TQR will be 75 right and not 45? otherwise it will not total up to 360. My question is that after finding the 4 angles, we can find QS by using cos 120 formula, but how will we calculate QR and RS from the given info?
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What is the perimeter of PQRS ? (1) x = 30 degree (2) w= 45 degree 21 Jan 2025, 07:38
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