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

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}

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.

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.

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

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.

hi everyone ,

this was a good question and very well explained by blueseas. I had only one question as blueseas himself states


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.

Therefore the condition would be:

If we have a triangle which sides have respectively lenghts of: 1 , 2 and X . And if one of the angles is 60 than it is a right triangle with the ratio 1:2: root of 3

Am i right here?

No. A triangle is uniquely defined by 2 sides and the INCLUDED angle. Not any angle. If two sides are 1 and 2, the triangles will be different depending on which angle is 60 degrees. If the angle between the sides of 1 and 2 length is 60, it will be a right triangle. Else it will not be right triangle.

Note that Q - X is just the angle TQR.
Angle Y is 180 -60 = 120 degrees.

All angles of the quadrilateral QRST add up to 360 = TQR + 120 + 45 + 120
So TQR = 75 degrees = Q - X

I used the assumption from begining that QR and PS are paralell (looking at the diagram), and got to A .... but reading the posts made it clear that it was a false assumption. Is there any situation we can trust the illustration or it is there only to mislead? or a fact like that should be always stated?

Do not draw inferences based on the diagrams. Perhaps the parallel lines have a slight slant that you are unable to make out. Perhaps the angle that seems 90 is actually 95 degrees you can't say. Also, the figures are to scale given the information in the question stem. They may not be to scale given the information in the 2 statements.

VeritasPrepKarishma

If in a quadrilateral if one pair of opposite angles are equal then can we conclude that it will be a parallelogram?
In this example Y = 120 and QRS = 120. Can we conclude that QRST is a parallelogram?

No, you cannot. Look at this diagram:


from
https://commons.wikimedia.org/wiki/File ... square.svg

Note the Kite has east and west angles equal but it is not a parallelogram.

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 = 75 degrees.

So the angles are 75, 75, 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

Hello, can you elaborate in st.2 how do we take for granted that Q=45 degrees?
I mean we do not know that Q=W or that R=Y. It is not stated that PS is a straight line, how can we assume that Y=180-60=120?

Lines that look straight in diagrams are straight.
Anyway, you are asked the perimeter of PQRS so PS must be a straight line. Else, the question would have asked for the perimeter of the figure PQRST.

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