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In the figure above, PQRT is a rectangle. What is the length of segmen  [#permalink]

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Difficulty:   5% (low)

Question Stats: 84% (01:37) correct 16% (01:32) wrong based on 335 sessions

### HideShow timer Statistics 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.

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Attachment: 2019-04-26_1413.png [ 4.76 KiB | Viewed 3092 times ]

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Re: In the figure above, PQRT is a rectangle. What is the length of segmen  [#permalink]

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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.
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In the figure above, PQRT is a rectangle. What is the length of segmen  [#permalink]

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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.

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Originally posted by DavidTutorexamPAL on 27 Apr 2019, 07:50.
Last edited by DavidTutorexamPAL on 09 May 2019, 10:59, edited 1 time in total.
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Re: In the figure above, PQRT is a rectangle. What is the length of segmen  [#permalink]

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Top Contributor
Bunuel wrote: 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.
Attachment:
2019-04-26_1413.png

Let's assign some variables to some of the lengths... 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 + z)/2 = 39
Replace z (aka TS) with 6 to get: (x)(y + 6)/2 = 39
Multiply both sides by 2 to get: (x)(y + 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

Cheers,
Brent

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Re: In the figure above, PQRT is a rectangle. What is the length of segmen  [#permalink]

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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

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_________________ Re: In the figure above, PQRT is a rectangle. What is the length of segmen   [#permalink] 10 May 2019, 16:10
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