In the figure above, PQRT is a rectangle. What is the length of segmen

Question

https://gmatclub.com/forum/download/file.php?id=47592 In the figure above, PQRT is a rectangle. What is the length of segment PQ ? (1) The area of region PQRS is 39 and TS = 6. (2) The area of region PQRT is 30 and QR = 10. DS75602.01 OG2020 NEW QUESTION 2019-04-26_1413.png

BrentGMATPrepNow · Accepted Answer

Let's assign some variables to some of the lengths...
 https://i.imgur.com/BpaiRKm.png

Target question :    What is the value of x?

Statement 1 : The area of region PQRS is 39 and TS = 6.  
Region PQRS is a TRAPEZOID

Area of trapezoid = (height)(base1 + base2)/2 
So, we get: (  x  ) y  + ( y + z) ]/2 = 39

Replace z (aka TS) with 6 to get: (x)(y + y + 6)/2 = 39
Simplify to get: (  x  )(2y + 6)/2 = 39

Multiply both sides by 2 to get: (  x  )(2y + 6) = 78
At this point, we can see that there's no way to definitively solve this equation for  x (aka PQ)  
Statement 1 is NOT SUFFICIENT

Statement 2 : The area of region PQRT is 30 and QR = 10 
PQRT is a rectangle, so the  area = (base)(height) 
We can write: xy = 30
Replace y (aka QR) with 10 to get: (x)(10) = 30
Solve:  x = 3 (i.e., PQ = 3) 
Since we can answer the  target question  with certainty, statement 2 is SUFFICIENT

Answer: B

Cheers,
Brent

RELATED VIDEO 
 https://www.youtube.com/watch?v=uQ356PWGq98

IanStewart · Answer

We can do everything algebraically: If we use W and L to represent the width and length of the rectangle, so W is the length of PQ (which we want to find), and if we use B to represent the line TS, which is the base of the triangle, then the area of the rectangle is LW, and the area of the triangle is BW/2.

Statement 1 tells us that if we add the rectangle's area and the triangle's area, we get 39. It also tells us that B = 6. So

LW + (BW/2) = 39
LW + (6W/2) = 39
LW + 3W = 39
W(L+3) = 39

So we might have W = 1 and L = 36, say, or W = 3 and L = 10, among other possibilities.

But there's a much faster way to see that Statement 1 is not sufficient. If you just draw the triangle's base TS of length 6, and draw some height RT that isn't too long (so the area of the triangle alone is less than 39), if we then start extending a rectangle to the left by drawing PT and QR, the area of the entire shape will get larger and larger the longer we make the rectangle's length. For exactly one length of the rectangle, the area of the entire shape will be exactly 39. But that length will be different for any height we choose - the length will need to be big if the height is small, and will need to be small if the height is big, so we can't possibly find the dimensions of the figure.

Statement 2 tells us LW = 30 and L = 10, so W = 3 and Statement 2 is sufficient. The answer is B.

DavidTutorexamPAL · Answer

A quick look at the statements shows that they relate to areas. When it comes to rectangles, in order to find the length of a side we'll need, apart from the area,the length of the other side. As for the right triangle, if we know one leg we'll need the other one.
Statement (1) gives us a region made of the rectangle and the right triangle, and thus knowing the total area and one leg does not suffice (remember - in a rectangle you need at least one side in addition to the area).
Statement (2) is definitely sufficient, since it give us the rectangular area and the other side of the rectangle. 
The correct answer is (B).

Posted from my mobile device

EMPOWERgmatRichC · Answer

Hi All,

We're told that PQRT is a rectangle. We're asked fo the length of segment PQ. This question can be solved with a mix of Geometry rules and TESTing VALUES. To start, when dealing with complex shapes, it helps to break the shape down into 'pieces': here, we're dealing with a rectangle and a right triangle.

(1) The area of region PQRS is 39 and TS = 6.

TS is the 'base' of the triangle, but we don't know anything about either the height/width of the triangle/rectangle or the length of the rectangle. Thus, there are lots of possible values for PQ. Here are two examples: 
IF.... 
PQ = 1, then the area of the triangle is (1/2)(6)(1) = 3 and the area of the rectangle is 39 - 3 = 36. This makes the length 36 and the answer is 1. 
PQ = 2, then the area of the triangle is (1/2)(6)(2) = 6 and the area of the rectangle is 39 - 6 = 33. This makes the length 16.5 and the answer is 2. 
Fact 1 is INSUFFICIENT

(2) The area of region PQRT is 30 and QR = 10.

The information in Fact 2 gives us the length of the rectangle and the area of the rectangle, so we can solve for the width: 
Area = (Base)(Height) 
30 = (10)(Height) 
3 = Height 
Fact 2 is SUFFICIENT

Final Answer:  B

GMAT assassins aren't born, they're made, 
Rich

mrcentauri · Answer

We are not dealing with a right triangle - there is no way to infer this from the image.

All we know is that the rectangle has 2 equal sides 2 equal angles, and 4 x 90 degree angles

the triangle could be any dimension - we can't infer this

dave13 · Answer

mrcentauri IMO actually its right triangle, cause width of rectangle is perpendicular to its length, forming 90 degrees angle

avigutman · Answer

Video solution from Quant Reasoning: Subscribe for more: https://www.youtube.com/QuantReasoning? ... irmation=1 https://www.youtube.com/watch?v=pgP2laPBhCo

ScottTargetTestPrep · Answer

Solution:

Question Stem Analysis:

We need to determine the length of segment PQ. Notice that it’s the height of trapezoid PQRS.

Statement One Alone:

Since we don’t know the length of PT or QR (except that they are equal in length), we can’t determine PQ. For example, if PT = QR = 2, then the lower base PS of trapezoid PQRS is 8. So PQ = 39/  = 39/5 = 7.8. However, if PT = QR = 3, then the lower base PS of trapezoid PQRS is 9. So PQ = 39/  = 39/6 = 6.5. Statement one alone is not sufficient.
 
 Statement Two Alone:

Since PQRT is a rectangle and QR (the length of rectangle PQRT) is 10, then the width of rectangle PQRT is PQ, and PQ = 30/10 = 3. Statement two alone is sufficient.

Answer: B

GMATinsight · Answer

Answer: Option B

Video solution by  GMATinsight

https://www.youtube.com/watch?v=zLzmesP7c7U

kop18 · Answer

Hi
I have a conceptual doubt with this question. 
Let's say the area of PQRS = area of rectangle PQRT + area of right angle triangle TRS.
Let's take the length and breadth of rectangle PQRT to be x and y (y is also the height for triangle for TRS) and m the base of the triangle TRS.

Then we have, 
xy+ (my)/2= 39......A

From statement 1, we know m=6. when we substitute this value in eqn A we get y(x+3)=39 (13*3). 
I think that statement 1 is sufficient because if 13*3 = y (x+3) then it means that either :
A. y=3 and x+3 = 13, or
B. y=13 and x+3=3
(I have done this because this is an equality with product of 2 primes and should ideally not have any other possibilities)

Option B is not possible as x cannot equal 0. Since 39 is the product of 2 prime factors we are able to deduce a definite value of x in this question.

Please can  Bunuel   GMATNinja   Brent   mike   scott  help me out with this? I am not sure where I am going wrong when I am applying the equality?

IanStewart · Answer

If you see an equation like ab = 39, where a and b must be positive integers, then it's certainly true that a and b both need to be divisors of 39, but there will actually be four different values a (or b) can have: 1, 3, 13 and 39.

In this question, if x and y need to be positive integers, and (x+3)(y) = 39, then x+3 can still equal 13 and 39, so there are two integer solutions. But we also don't even know that x and y are integers, so there are infinitely many solutions, and we can't use divisibility to think about the equation here.

kop18 · Answer

Thank you very much  Ian  - this was helpful. I was only thinking of the dimensions in terms of integers but I agree with your explanation that these dimensions can be non-integers and hence, there can be infinite possibilities.

In the figure above, PQRT is a rectangle. What is the length of segmen

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In the figure above, PQRT is a rectangle. What is the length of segmen

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