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

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.
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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.