In rectangular region PQRS above, T is a point on side PS. If PS = 4,

VERITAS PREP OFFICIAL SOLUTION:

Question Type: What Is the Value? This question asks for the area of the rectangle PQRS.

Given information in the question stem or diagram: The diagram is given and importantly one side is defined with PS = 4, so both PS and QR = 4. To find the area you need to somehow determine either RS or PQ from the information in the statements. The question is really: What is RS or PQ?

Statement 1: QTR is an equilateral triangle. If this is true, then with QR as the base, the height of the equilateral triangle is equivalent to the side that you are trying to determine (PQ or RS). Since the base of QTR is 4 you can determine the necessary height with your knowledge of 30–60–90 triangles or simply with your knowledge that if you know one thing about an equilateral triangle, then you know everything! While you do not need to calculate it, the height would be 2√3 as it is the long leg in a 30–60–90 triangle formed by 1/2 the base of QTR (2), the hypotenuse QT (4) and the height ( 2√3 ) which is equivalent to PQ or RS. Statement 1 is sufficient, so the answer is A or D.

Statement 2: In this difficult statement, the testmakers are playing with a common trick. They have polluted your brain with the first statement and want you to assume that if “segments PT and TS have equal lengths,” then QTR must again be equilateral. However, this statement does nothing to help you determine the length of the sides (QP and RS) because it only proves that QTR is isosceles. There is no limit put on the lengths of PQ and RS (because you do not know the angle of TQR and TRQ) with this statement, so it is not sufficient. Remember: One of the keys to success in Data Sufficiency is to consciously avoid assumptions, but that can be hard when you are set up so nicely to make assumptions with the other statement. Statement 2 is not sufficient, so the correct answer is A.
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(1) QTR is equilateral means that point T is in the middle of PS - > qt=rt=long leg of the eq.triangle=4 and than from the pytgorean theorem we can find QP and the Area - SUFFICIENT
(2) Smae Info as above SUFFICIENT
Answer (D)

second statement is not sufficient. B'se QT may be equal to RT, but need not be equal to QR. so 2 is insufficient
So i believe answer will be 'A'

1. PS=QR=4
Since , QTR is equilateral
QT=TR=QR=4
Triangles QTP and RTS are congruent(RHS)
=>PT=TS
Now using pythagorean theorem , we can find out QP or RS .
QT^2= QP^2 + PT^2
And calculate the area of rectangle PQRS.
Sufficient

2. PT=TS = 2
Triangles QTP and RTS are congruent (SAS)
=> QT=RT
But we have no can conclude whether QT/RT is equal to QR .
Hence, insufficient.

Answer A

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Hmm... was too fast about B, missed the point... B is not sufficient

SOLUTION:

Statement 1: If Triangle QTR is equilateral. then QR=QT=4, And also every angle of this Triangle is 60 degrees.
then Triangle QTP is 30-60-90 right triangle then the sides of triangle PQT are in ratio 1:\(\sqrt{3}\):2.

if longest side QT=4, then PQ must be 2*\(\sqrt{3}\).

we now know lengths of adjacent sides , we can calculate area of rectangle.

So , Statement 1 is SUFFICIENT.

Statement 2: we know PT, but we don't know anything about type of a triangle or lengths of PQ or QT.

So, Statement 2 is INSUFFICIENT.

ANS: A.

Hope i am not missing anything about Statement 2.

The correct answer must be D
If PT = PS , then we already know QP=RS , Hence QT = RT . Therefore triangles QPT and RTS are congruent (RHS) .
Angles TQR and QRT are equal ( As QT=RT) .
Angles QTP and RTS are equal ( Congruent triangles) .
Angle QTR = 180 - 2(angle TQR) = 180 - 2(angle QTP)
Therefore angle TQR = QTP , which implies TQR=QTP=RTS
Therefore an equilateral triangle , Hence we can find the area .

The correct answer is given under the spoiler in the original post and it's A. For (2) all we k now is one side of the rectangle. Having T as the midpoint of PS gives us nothing.

Hope it helps.
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Yes i have seen the answer .
But i proved that if T is the midpoint then the resulting triangle will be equilateral .

Hi,
check the angles which are equal..
the Triangle is QRT
and your solution says

The angle which has to be shown EQUAL are QRT = RTQ = TQR..
Statement 2 is not sufficient..
relook in your solution

Logically, I will give you ONE rectangle..
1) It has 100 as length and ONLY 1 as width..
try it out, it can never have all angles as 60 degree as the two equal sides will be CLOSE to 50-60 and the third side will be 100..
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I got this right but I'm curios how it would help us if we find the area of QTR to find the area of the rectangle.
can some one address the query?

For an equilateral triangle, all the angles are equal to 60 degree.
So ag(QRT)+ag(TRS)=90
ag(TRS)= 30degree

which means: ag(RTS) = 60 and ag(RST)=90

Sin 60 = root3/2

RS/RT=root3/2

RS=2root3


area= RS*PS

1 suff

(1) PQ=PS=4
Since QTR is equilateral then, knowing the value of one side we can easility calculate the altitude,using the followign formula:
h=a*3^1/2/2=4*3^1/2/2=2*3^1/2 Sufficient
(2) Defferent values are possible and since we are lacking any information about angles it's completely insufficient.

OFFICIAL EXPLANATION:

It is given that PQRS is a rectangle and PS = 4. The area of PQRS can be determined if and only if PQ can be determined.

1) It is given that ΔQTR is equilateral. If 62823.png is the height of ΔQTR as shown above, then ΔQUT is a 30−60−90° triangle with UR = 2. Using the ratios for 30−60−90 triangles, TU = 62832.png. Since PQ = TU, the area of PQRS can be determined; SUFFICIENT.
 
2) Given that PT = TS, PQ could be any positive number. Thus, it is not possible to determine PQ and therefore not possible to determine the area of PQRS; NOT sufficient.

ANswer: Option A

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