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Math Expert V
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In the figure above, what is the area of region PQRST ? (1) PQ = RS (  [#permalink]

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17 00:00

Difficulty:   65% (hard)

Question Stats: 62% (02:06) correct 38% (01:53) wrong based on 672 sessions

### HideShow timer Statistics In the figure above, what is the area of region PQRST ?

(1) PQ = RS
(2) PT = QT

DS74602.01
OG2020 NEW QUESTION

Attachment: 2019-04-26_1406.png [ 9.92 KiB | Viewed 5979 times ]

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In the figure above, what is the area of region PQRST ? (1) PQ = RS (  [#permalink]

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The Logical approach to this question will the observation that the area we're looking for is of a shape which doesn't have a matching formula. Thus, we should either refer to it as the sum of or the difference between two familiar shapes. In this case, the sum of a rectangle and a triangle.
Statement (1) tells us that the sides of the rectangle are 6, which is enough to find the other side of the rectangle (using the pythagorean triplet 6,8,10). However, we don't have a height required to find the area of the triangle. Thus, answers (A) and (D) are eliminated.
Statement (2): Knowing that the triangle is an isosceles triangle is not enough to find its area. Answer choice (B) is eliminated.
Combining both statements, we now know that the triangle is an equilateral triangle, so the length of one side is enough to find its area.

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Originally posted by DavidTutorexamPAL on 26 Apr 2019, 13:56.
Last edited by DavidTutorexamPAL on 04 May 2019, 09:42, edited 1 time in total.
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Re: In the figure above, what is the area of region PQRST ? (1) PQ = RS (  [#permalink]

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Please explain why statement 2 is insufficient:
Since we know the isosceles triangle. Side QT = RS, and Side QR can be found using Pythagorean theorem. Hence, it can be found. Thanks.
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In the figure above, what is the area of region PQRST ? (1) PQ = RS (  [#permalink]

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Valhalla wrote:
Please explain why statement 2 is insufficient:
Since we know the isosceles triangle. Side QT = RS, and Side QR can be found using Pythagorean theorem. Hence, it can be found. Thanks.

Hey Valhalla,

Knowing that QT= RS is insufficient to know their length. (In particular, it is also insufficient to calculate QR)
Try it!
Draw PT of length 6, build an isosceles triangle on top of it and build the square QRST off the side of that.
See how changing the lengths of the sides of the isosceles means that you will change the size of the square (because the diagonal must equal 10).
Then this is insufficient.
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Re: In the figure above, what is the area of region PQRST ? (1) PQ = RS (  [#permalink]

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Thanks. I have mistakenly taken PT = QT = 6, that is why I could not fathom how statement 2 alone is insufficient. Dear David, thanks for replying. Any pointers you would give to avoid such mistakes? Thanks.
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In the figure above, what is the area of region PQRST ? (1) PQ = RS (  [#permalink]

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Valhalla wrote:
Thanks. I have mistakenly taken PT = QT = 6, that is why I could not fathom how statement 2 alone is insufficient. Dear David, thanks for replying. Any pointers you would give to avoid such mistakes? Thanks.

Hey Valhalla,

The mistake you describe is what we classify as a 'silly mistake'.
In other words, it is not that you didn't know the material, but rather that you missed / misinterpreted a small piece of data and therefore got the rest of the question wrong.
Tip number 1: Take a few seconds after reading the question but before diving into calculations to make sure you've copied everything down / read everything correctly. This is the best way to avoid such mistakes.
Tip number 2: Build up a timing strategy which doesn't leave you feeling completely rushed. It is better to guess a few questions and have enough time to do the others properly than to do everything in a hurried manner.
Tip number 3: Practice! The more questions you solve, the more familiar with the GMAT you get, and consequently the better you are able to notice the 'important' and 'tricky' details.

Best of luck,
David
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Re: In the figure above, what is the area of region PQRST ? (1) PQ = RS (  [#permalink]

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1
Why is statement 1 insufficient?
We have diagonal as 10 and PQ=RS=6
So, we can find ST = 8
Since its a rectangle, we can directly find the area of this figure.
Similarly, for the triangle, we can take height which is divides QT in half and so by pythagoras, we can find the height of triangle as 5.

What have I considered wrong here?
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Re: In the figure above, what is the area of region PQRST ? (1) PQ = RS (  [#permalink]

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abannore wrote:
Why is statement 1 insufficient?
We have diagonal as 10 and PQ=RS=6
So, we can find ST = 8
Since its a rectangle, we can directly find the area of this figure.
Similarly, for the triangle, we can take height which is divides QT in half and so by pythagoras, we can find the height of triangle as 5.

What have I considered wrong here?

Hey abannore,

The height only divides QT in half if the triangle is isosceles (if PQ = PT).
One way to SEE that there are many options is to imagine different angles for PQT.
If angle PQT were exactly 90 degrees, then PQ itself would be the height!
Instead, if angle PQT was very close to 0 then the triangle would be very 'narrow', and the height would also be very close to 0.
Two different options --> insufficient.

Best,
David
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Re: In the figure above, what is the area of region PQRST ? (1) PQ = RS (  [#permalink]

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DavidTutorexamPAL wrote:
The Logical approach to this question will the observation that the area we're looking for us of a shape which doesn't have a matching formula. Thus, we should either refer to it as the sum of or the difference between two familiar shapes. In this case, the sum of a rectangle and a triangle.
Statement (1) tells us that the sides of the rectangle are 6, which is enough to find the other side of the rectangle (using the pythagorean triplet 6,8,10).Yet, we don't have a height required to find the area of the rectangle. Thus, answers (A) and (D) are eliminated.
Statement (2): Knowing that the triangle is an isoscelous triangle is not enough to find its area. Answer choice (B) is eliminated.
Combining both statements, we now know that the triangle is an equilateral triangle, so one side is enough in order to find its area.

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Hi David, do you mean triangle here?
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In the figure above, what is the area of region PQRST ? (1) PQ = RS (  [#permalink]

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rnn wrote:
DavidTutorexamPAL wrote:
The Logical approach to this question will the observation that the area we're looking for us of a shape which doesn't have a matching formula. Thus, we should either refer to it as the sum of or the difference between two familiar shapes. In this case, the sum of a rectangle and a triangle.
Statement (1) tells us that the sides of the rectangle are 6, which is enough to find the other side of the rectangle (using the pythagorean triplet 6,8,10).Yet, we don't have a height required to find the area of the rectangle. Thus, answers (A) and (D) are eliminated.
Statement (2): Knowing that the triangle is an isoscelous triangle is not enough to find its area. Answer choice (B) is eliminated.
Combining both statements, we now know that the triangle is an equilateral triangle, so one side is enough in order to find its area.

Posted from my mobile device

Hi David, do you mean triangle here?

Hey rnn,
You mean in (1)?
Yes! Thanks, fixed the typo.
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Re: In the figure above, what is the area of region PQRST ? (1) PQ = RS (  [#permalink]

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Hi All,

We're asked for the area of region PQRST. This question is based around a couple of Geometry rules. To properly find the area, we can break this shape into 2 pieces (a rectangle and a triangle), so we need to know the dimensions of the rectangle and the exact type of triangle - since we have one of the sides, we either need the 3 angles or the 2 missing sides - to calculate the overall area.

(1) PQ = RS

With the information in Fact 1, we know that the width of the rectangle is 6 and one of the two missing triangle sides is also 6. Along with the diagonal of the rectangle, we can now calculate its length (it's 8, since we have a 6/8/10 right triangle in the rectangle) but without the 3rd side of the triangle (or its 3 angles), we cannot calculate that other area.
Fact 1 is INSUFFICIENT

(2) PT = QT

The information in Fact 2 tells us that the triangle is either Isosceles or Equilateral, but we still do not know enough to determine its area and we don't know the length or width of the rectangle, so we cannot determine its area either.
Fact 2 is INSUFFICIENT

Combined, we know...
PQ = RS
PT = QT

With both Facts, we know that the triangle's sides are 6/6/6, so it's Equilateral and we can calculate its area. We also know the area of the rectangle (it's 6x8 = 48), so we CAN determine the area of PQRST.
Combined, SUFFICIENT

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Re: In the figure above, what is the area of region PQRST ? (1) PQ = RS (  [#permalink]

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Hi everyone.

I have a “silly ques”- why cant this alleged rectangle be a square? What property am I missing?

TIA

Sambhav

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Re: In the figure above, what is the area of region PQRST ? (1) PQ = RS (  [#permalink]

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DavidTutorexamPAL wrote:
abannore wrote:
Why is statement 1 insufficient?
We have diagonal as 10 and PQ=RS=6
So, we can find ST = 8
Since its a rectangle, we can directly find the area of this figure.
Similarly, for the triangle, we can take height which is divides QT in half and so by pythagoras, we can find the height of triangle as 5.

What have I considered wrong here?

Hey abannore,

The height only divides QT in half if the triangle is isosceles (if PQ = PT).
One way to SEE that there are many options is to imagine different angles for PQT.
If angle PQT were exactly 90 degrees, then PQ itself would be the height!
Instead, if angle PQT was very close to 0 then the triangle would be very 'narrow', and the height would also be very close to 0.
Two different options --> insufficient.

Best,
David

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Re: In the figure above, what is the area of region PQRST ? (1) PQ = RS (  [#permalink]

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Sam10smart wrote:
Hi everyone.

I have a “silly ques”- why cant this alleged rectangle be a square? What property am I missing?

TIA

Sambhav

Posted from my mobile device

Hi Sambhav,

In my explanation (which is right above yours), I work through that the information in Fact 1 proves that the rectangle is a 6x8 (and not a square). I didn't go through that extra work in Fact 2 because it wasn't necessary - BUT the information in Fact 2 does not restrict the rectangle to a 6x8. With Fact 2, we could actually be dealing with a square (if the sides were 5√2, then we would have a square and an isosceles triangle (with a different total area than if we had a 6x8 rectangle and triangle).

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Re: In the figure above, what is the area of region PQRST ? (1) PQ = RS (  [#permalink]

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Hello All,

How can we came to conclusion that QRST is rectangle? Does four angles =90 degree is sufficient condition for a figure to be
rectangle? As we don't have any information that opposite sides are equal or parallel

Thanks
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Re: In the figure above, what is the area of region PQRST ? (1) PQ = RS (  [#permalink]

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Hi anand.prakash90,

Yes - when dealing with a 4-sided shape, if all four angles are 90-degree angles, then the sides ARE parallel and the shape IS a rectangle (and might actually be a square, which is ultimately a specific type of rectangle).

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Re: In the figure above, what is the area of region PQRST ? (1) PQ = RS (  [#permalink]

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Top Contributor
Bunuel wrote: In the figure above, what is the area of region PQRST ?

(1) PQ = RS
(2) PT = QT
Attachment:
2019-04-26_1406.png

Target question: What is the area of region PQRST ?

Statement 1: PQ = RS

IMPORTANT: For geometry Data Sufficiency questions, we are typically checking to see whether the statements "lock" a particular angle, length, or shape into having just one possible measurement. This concept is discussed in much greater detail in the video below.

This technique can save a lot of time.

Now that we know PQ = RS, we know that RS = 6
Since the diagonal RT = 10, we can apply the Pythagorean Theorem to see that QR = 8, which means rectangle QRST has area 48
HOWEVER, we don't yet know the area of triangle PQT. Here's why:
Notice that no given information LOCKS IN the position of side PQ.
If we think of side PQ as a door with its hinge at point Q, then side PQ can freely move around, changing the area of triangle PQT
Since triangle PQT can have a variety of angles, we cannot determine the area of region PQRST
Since we cannot answer the target question with certainty, statement 1 is NOT SUFFICIENT

Statement 2: PT = QT
Since we aren't provided any information about the lengths of PT and QT, we cannot determine the area of region PQRST
Since we cannot answer the target question with certainty, statement 2 is NOT SUFFICIENT

Statements 1 and 2 combined
Statement 1 tells us that rectangle QRST has area 48
From statement 1, we also see that side QT = 6

Statement 2 tells us that PT = QT = 6
So, triangle PQT is an EQUILATERAL with sides of length 6.
So, we now have enough information to find the area of PQRST.
Since we can answer the target question with certainty, the combined statements are SUFFICIENT

Cheers,
Brent

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