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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.
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 [ 23.71 KiB | Viewed 5172 times ]
29 Jun 2017, 02:53
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msonnie17
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 5221 times ]
Quote:
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
23 Nov 2017, 04:38
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VeritasPrepKarishma
Responding to a pm:
Quote:
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.
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
)
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
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
Answer: B
