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

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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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Re: 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]
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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Re: 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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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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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

Posted from my mobile device
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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]
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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Genoa2000 wrote:
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.

Could you please explain how would you have found the area with knowing the 3 angles? (If these angles were not 30-60-90 or 45-45-90). Thanks!

Hi Genoa2000,

If you know any one of the sides in a 30/60/90 or 45/45/90 right triangle, then you can figure out all of the other sides (and then the area, perimeter, etc.). The reason why is because there are math 'relationships' between angles and the lengths of the sides that are across from those angles.

With those two types of triangles, we know already know the relationships (and you probably already have them memorized): 30/60/90 = X/X√3/2X and 45/45/90 = Y/Y/Y√2

For example, if a side length of 6 is across from the 30 degree angle in a 30/60/90, then we know that the 3 sides are 6/6√3/12.

With other triangles, there are still relationships, but we would need Trigonometry to define those exact relationships - they ARE defined though (and in a DS question, if you know that there is only one answer to the given question, then you have a 'Sufficient' situation (regardless of whether you can actually do the math to define that answer or not).

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

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

Solution:

Question Stem Analysis:

We need to determine the area of region PQRST. Notice that the area of the region is composed of triangle PQT and rectangle QRST.

Statement One Alone:

From statement one, we see that RS = 6. Since triangle RST is a right triangle with side RS = 6 and hypotenuse RT = 10, side ST = 8 (notice triangle RST is a 6-8-10 right triangle). Since RS and ST are also the sides of rectangle QRST, the area of rectangle QRST is 6 x 8 = 48. However, we can’t determine the area of triangle PQT, so we can’t determine the area of region PQRST. Statement one alone is not sufficient.

Statement Two Alone:

From statement two, we see that triangle PQT is at least an isosceles triangle and perhaps an equilateral triangle. However, since we don’t know which one it really is, we can’t determine its area. Statement two alone is not sufficient.

Statements One and Two Together:

With the two statements, we see that RS = QT and since RS = 6, QT = 6. Furthermore, PT = QT = PQ = 6. This makes triangle PQT an equilateral triangle. Since we know a side of the equilateral triangle, we can determine its area. Lastly, since we’ve already determined the area of rectangle QRST to be 48, we can determine the area of region PQRST. Both statements together are 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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Bunuel wrote:

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

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Re: In the figure above, what is the area of region PQRST ? (1) PQ = RS ( [#permalink]
I do not understand how statement two is not sufficient. We have square with a diagonal that equals 10. Diagonal inside a square is equal to S√ 2, Where S is equal to the side of the square. 10=S√ 2. Divide by root. We now have a value for S. With this value, we know that QT is equal to the value of S because QT is a side of the square. Knowing that PT=QT gives us our final value. What am I missing?

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Re: In the figure above, what is the area of region PQRST ? (1) PQ = RS ( [#permalink]
Hi Nmerritt1995,

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.

With Fact 2, we are told that PT = QT. This means 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. It might be a square, but it might also be a rectangle.
Fact 2 is INSUFFICIENT

It's not until we combine all of the information that we are given that we know for sure that the triangle is Equilateral and the rectangle is a 6 x 8.

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

If the diagonal bisects the rectangle, why can't it create two 45-45-90 triangles, thus making QT and QR 5\sqrt{2} ? Please let me know what I am missing.

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