A six-sided mosaic contains 24 triangular pieces of tile of the same

Question

https://gmatclub.com/forum/download/file.php?id=47588 A six-sided mosaic contains 24 triangular pieces of tile of the same size and shape, as shown in the figure above. If the sections of tile fit together perfectly, how many square centimeters of tile are in the mosaic? (1) Each side of each triangular piece of tile is 9 centimeters long. (2) The mosaic can be put inside a rectangular frame that is 40 centimeters wide. DS24602.01 OG2020 NEW QUESTION 2019-04-26_1400.png

DavidTutorexamPAL · Accepted Answer

The Logical approach to this question will focus on looking for the length of the side of one tile, since one side is enough in order to find the area of an equilateral triangle.
Statement (1) gives us just what we were looking for. Answer choices (B), (C) and (E) are eliminated.
Statement (2) doesn’t an exact measure, and as we're not told whether it is inscribed or not. Answer choice (D) is also eliminated.
The correct answer is (A).

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ZoltanBP · Answer

Let  A  be the area of the mosaic. The original question:  A=?

1) We know that the small tiles are equilateral triangles with side lengths of 9. Since the formula for the height of an equilateral triangle is  height=\frac{\sqrt{3}}{2}\cdot base , the area of the mosaic is the following.

A=24\cdot \frac{9\cdot \frac{\sqrt{3}}{2}\cdot 9}{2}

Thus, the answer to the original question is a unique value.  \implies   Sufficient

2) We know that the frame in which the mosaic is placed is 40 cm wide, but no information is given about the side lengths of the 24 identical triangles. Even if the bases of these identical isosceles triangles were given, different heights would give us different areas. Thus, we can't get a unique value to answer the original question.  \implies   Insufficient

Answer: A

Archit3110 · Answer

#1
hexagon is 6 times equilateral ∆ area
given side = 9 sufficeint to solve
#2
mosaic rectangluar size ; insufficeint
IMO A

EMPOWERgmatRichC · Answer

Hi All,

We're told that a six-sided mosaic contains 24 triangular pieces of tile of the SAME size and shape, as shown in the figure above and that the sections of tile fit together perfectly. We're asked for the total square centimeters of tile in the mosaic. While this question might look a bit 'crazy', it's a great "concept question" meaning that if you know the concepts involved, then you won't have to do much math to actually get the correct answer. To answer this question, we'll need information to determine the area of any one of the triangles, then we can multiply that area by 24 to get the total area of the mosaic.

1) Each side of each triangular piece of tile is 9 centimeters long.

Fact 1 tells us that we're dealing with Equilateral triangles with sides of 9 cm. With this information, we can determine the area of each triangular tile (the best part is that we don't actually have to do that math here; we know that we CAN do that math - and that there will be just one value for the area - so we can stop working). 
Fact 1 is SUFFICIENT

2) The mosaic can be put inside a rectangular frame that is 40 centimeters wide.

The information in Fact 2 puts a limit on what the maximum side of each triangle could be, but doesn't given us any data on the exact areas involved, so since the triangles don't have a set area, the answer to the question will change as the areas change. 
Fact 2 is INSUFFICIENT

Final Answer:  A

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

Sarjaria84 · Answer

ZoltanBP   DavidTutorexamPAL   GMATPrepNow

The question stem states "24 triangular pieces of tile of the same size and shape" so I am not sure why would we have different heights. Can we consider these small triangles to be 24 equilateral triangles and if the statement 2 would have stated that the mosaic shape is inscribed in the rectangular frame, then we would have had the height and subsequently we could get the side of the equilateral triangle and the area.

So the only missing info is whether the mosaic shape is inscribed or not. Is my thinking correct ?

Thanks
Saurabh Arjaria

altairahmad · Answer

GMATBusters   Bunuel   chetan2u 
Is it ok to say that STatement 2 doesn't tell us what kind these 24 triangles are. It only fixes the height of the triangles but unless we know something more about the triangles e.g equilateral, isosceles etc, we can determine the area of the individual triangle.

EMPOWERgmatRichC · Answer

Hi altairahmad,

You are correct that Fact 2 does NOT tell us what 'type' of triangles we're dealing with nor the actual measurements of any of the sides (that information only gives us a 'maximum base length'; since we know that the length of 4 'bases' will fit inside a 40 cm width, thus each 'base' is something LESS than 10cm).
 
GMAT assassins aren't born, they're made,
Rich

hemantbafna · Answer

(1) From the side of the equilateral triangle, we can get the value of the area of each triangle. 
Area of equilateral triangle= root 3/4*(side)^2

Area of the mosaic = root 3/4*(side)^2 * 24( Total number of triangular pieces) .  Sufficient .

(2) Nothing is known about the length, therefore.  Not Sufficient .

A

ScottTargetTestPrep · Answer

Solution:

Question Stem Analysis:

We need to determine the area of the mosaic, which is a hexagon comprised of 24 triangles of the same size and shape. Notice that if each triangle is an equilateral triangle and we are given a side of the triangle, then we can determine the area of one triangle and hence the areas of all 24 triangles, i.e., the area of the mosaic.

Statement One Alone:

Since we are given that each side of a triangular piece is 9 cm long, we know each triangle is an equilateral triangle. Therefore, we can determine the area of one triangle and hence the areas of all 24 triangles, i.e., the area of the mosaic. Statement one alone is sufficient.

Statement Two Alone:

Knowing the mosaic can be put inside a rectangular frame that is 40 centimeters wide is not sufficient to determine the area of the mosaic. The mosaic could be different sizes, as long it fits into the rectangular frame. However, since it can be different sizes, it can have different areas. Statement two alone is not sufficient.

Answer: A

Nidhijain94 · Answer

Bunuel   chetan2u  I have the same doubt. other answers kept saying that this statement is insufficient because the length of a rectangle is not mentioned. But my understanding is its length is not required. If mosaic fits, then one of the diagonal of mosaic will be equal to 40cm and we can find the side of an equilateral triangle. The only missing info is whether the mosaics diagonal fits the width of the rectangle. could you please help with your explanation?

SiddharthR · Answer

The question stem states "24 triangular pieces of tile of the same size and shape" so I am not sure why would we have different heights. Can we consider these small triangles to be 24 equilateral triangles and if the statement 2 would have stated that the mosaic shape is inscribed in the rectangular frame, then we would have had the height and subsequently we could get the side of the equilateral triangle and the area.

So the only missing info is whether the mosaic shape is inscribed or not. Is my thinking correct ?

Thanks
Saurabh Arjaria

Bunuel   chetan2u  I have the same doubt. other answers kept saying that this statement is insufficient because the length of a rectangle is not mentioned. But my understanding is its length is not required. If mosaic fits, then one of the diagonal of mosaic will be equal to 40cm and we can find the side of an equilateral triangle. The only missing info is whether the mosaics diagonal fits the width of the rectangle. could you please help with your explanation?

I agree with these statements and believe they are correct. Since the stem clearly states that we are dealing with a triangle which has the "same shape and size", it wouldn't be wrong to consider all the 24 triangles in the hexagon as equilateral triangles.

I believe the issue with statement II is that we cannot figure out if the rectangular frame is "circumscribing" the hexagon as stated in the first response of this question.

I must agree that this is a weird question.

NischalSR · Answer

I think, even if we assume "put inside" means "inscribed" (which would be wrong) and that all triangles are equilateral, we actually do not know how the mosaic actually fits in the rectangle. The 40 cms can be the BASE of 4 triangles or the HEIGHT of the 4 triangles.

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Kimberly77 · Answer

Sarjaria84  Thanks for the explanation. Why are we dividing it by 2 twice? Thanks

Hoozan · Answer

Hi  EMPOWERgmatRichC

Please could you explain how from (2) we get a limit on the sides of a triangle? We are told that the mosaic can fit in a rectangle of 40cm. So we can say that:

1) The two triangles at the base of the hexagon will cover  at least  40 cm.
2) The four triangles (the height of the hexagon) will cover  at least  40 cm.

avigutman · Answer

Video solution from Quant Reasoning: Subscribe for more: https://www.youtube.com/QuantReasoning? ... irmation=1 https://www.youtube.com/watch?v=Bwm3RLn4l08

Elite097 · Answer

GMATinsight   EMPOWERgmatRichC  I do not get how would we get the answer if we knew that the mosaic fits perfectly in the rectangle. Even if it did, we do not know what kind of a triangle is inside- for all we know they one triangle could have all different lengths or it could be isoceles or many such shapes and thus would have number of possible areas with respect to the hexagon so why would it be sufficient if we had info that hexagon fits into rectangle  ScottTargetTestPrep

EMPOWERgmatRichC · Answer

Hi Elite097, The information in Fact 2 (about the hexagon fitting inside of a particular rectangle) is NOT enough to sufficiently answer the question (which is why the correct answer is neither B or D). From your post, I assume that you've worked through the prompt and read some of the explanations in this thread - and I think that you understand that Fact 2 is Insufficient. Are there any other aspects of this question that you'd like to discuss in more detail? GMAT assassins aren't born, they're made, Rich Contact Rich at: Rich.C@empowergmat.com

Elite097 · Answer

@empowergmtrichc No I have not understood and I basically have these following questions:
1. Can we say it is a regular polygon from the fact that all triangles are identical?
2. What does 'fit perfectly' in the question stem mean? If it means closing gaps, could these triangles inside also be scalene, acute, obtuse, right-angled, isosceles?
3. Statement 2- I am not sure- If let's say we were told that it can be inscribed in a rectangle, then could we say that St 2 is Sufficient. And please explain your answer too s to why yes/ no.
4. For statement 2, what additional info in the statement could help us consider st 2 sufficient? Also, besides that what other ways could be sufficient ex- could it be suffienct if we know lengths of one triangle along with the type of that triangle? 
5. Can this be one reason for rejecting st 2- even though hexagon can fit inside, we do not know exact dimensions of hexagon and it could be big/small and hence area of whole hexagon could vary?

Tanchat · Answer

Dear Experts,  KarishmaB   Bunuel   BrentGMATPrepNow  and  ScottTargetTestPrep  and others

1) Is it possible that other types of Triangle can form this six-sided mosaic?

2) in the statement two, if the first question is NO, do we actually require the length of triangle? We know the wide of rectangle and the mosaic can be put into the rectangle, we definitely know the size of the triangle.

3) According to the statement two, (2) The mosaic can be put inside a rectangular frame that is 40 centimeters wide.
can we imply that the mosaic perfectly fits into the rectangular? or it is possible that this rectangular may be larger than the mosaic --> so the mosaic can be put into the rectangular but we cannot define any size of the mosaic (or triangle)

Thank you in advance

A six-sided mosaic contains 24 triangular pieces of tile of the same

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A six-sided mosaic contains 24 triangular pieces of tile of the same

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