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In the figure above, what is the value of x + y ? [#permalink]

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29 Apr 2013, 15:13

2

This post received KUDOS

bhavinp wrote:

Answer is E

Here's a rough version of the image.

Statement 1 is insufficient. We know angle x is 70, but we know nothing about the placement of point D. The lower the placement of point D the greater the angle y, so y could really be any number of values. Remember we can't assume anything about the placement of a point, unless we are explicitly told about the placement in the question.

Statement 2 is insufficient. x and y could be any number of values. For instance x could be 50 degrees or 70 degrees. All we know is that the triangles are isosceles, but we know nothing about the angles.

Statements 1 and 2 together are insufficient. From statement 1 we know x is 70 and that angles BAC and BCA are both 55. But again we know nothing about the placement of point D. Point D could be very close to B in which case y would be close to 70 degrees or point D could be close to the segment AC in which case y would be close to 180 degrees.

I hope that helps! Bhavin

Maybe this is a silly question - I understood this problem but I have one issue with the answer explanation. Why do both angles have to be 55? Isoceles triangles must have at least two equivalent angles, right? Why couldn't the angles have been 70/70/40? Doesn't change the answer from E, but still im curious

Merging similar topics. As for your question: yes, ABC could be 70-70-40 triangle as well.

Bunuel and others, I feel this question has a significant gap --- is the original diagram drawn to scale or not?

Of course, on the GMAT, we know "Figures are drawn as accurately as possible. Exceptions will be noted." I discuss this in a blog post. http://magoosh.com/gmat/2012/gmat-trick ... -possible/ This is a crucial fact for students to keep in mind when interpreting GMAT diagrams.

IF this diagram is purporting to be drawn as accurate as possible, then it's an exceptionally poor diagram. That angle looks nothing like a 70 degree angle. If it's drawn to scale, though, we have to accept that the diagram is somewhat close to symmetrical, and therefore, the 70-70-40 triangle would not be possible.

IF the diagram is not drawn to scale, which I suspect was the intent of the author, that needs to be explicitly stated. Then, the 70-70-40 triangle would be possible. Any bilateral symmetry is out the window if it's not drawn to scale. Here's a scaled diagram of the figure with the 70-70-40 triangle.

Attachment:

isosceles triangles.JPG [ 25.57 KiB | Viewed 16483 times ]

I believe, either way, the answer would be (E). Nevertheless, I think this is a crucial issue for students to consider while analyzing the possibilities for a given diagram on the GMAT.

Thanks for the detailed response -- makes a lot of sense.

One point i'd like to bring up because I think that my original question wasn't phrased accurately:

1) If they give us that the height of ADC is 1/2 the height of ABC, doesn't that imply that the angle is doubled since AC is the same for both triangles? I meant to ask this and not bring in the "inscribed triangle" rule

Thanks!

Dear russ9 Absolutely and 100% NO!! I don't know how to be any more clear about this. You are laboring under a very powerful geometry misconception. There is no easy proportional relationship between distances and angles. If it were, trigonometry would be such a ridiculous easy topic that teachers could cover it entirely in a day toward the end of a geometry class. Instead, trigonometry is such a difficult topic that it requires almost a full year on its own. If you have never learned trigonometry, suffice to say there are all kinds of subtleties about the relationship of lengths and angles that you do not and cannot appreciate. That's OK. You don't need to know any of that for the GMAT. All you have to do is rid yourself of this pernicious misconception.

Consider this diagram:

Attachment:

Isosceles with half altitude.JPG [ 21.61 KiB | Viewed 1157 times ]

Triangle ABC is a 70-55-55 isosceles triangle. Point D is the midpoint of altitude BE, and the angle at D is not a nice even 140 degrees. Instead, it's an irrational angle, that is, an angle whose measure is an irrational number. This is very typically the case in trigonometry: for example, triangles with all integer sides must have at least two angles that have irrational measures, and triangles with integer angle measures almost always have at least two sides that have irrational lengths. Again, all this is much more than you need to know. I simply want to impress upon you that there are absolutely no magic proportions that link the behavior of lengths and the behavior of angles.

Re: In the figure above, what is the value of x + y ? [#permalink]

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29 Apr 2013, 16:51

Well, this is more of analytical question. When i looked at this question , i tried to remember a formula that i could use but unfortunately couldn't fit anything but looking at the image i see a very generic image. Their is no way to determine x + y. As infinite number of such triangles can be drawn, which will satisfy the given constraints. Hence E wins.

Statement 1 is insufficient. We know angle x is 70, but we know nothing about the placement of point D. The lower the placement of point D the greater the angle y, so y could really be any number of values. Remember we can't assume anything about the placement of a point, unless we are explicitly told about the placement in the question.

Statement 2 is insufficient. x and y could be any number of values. For instance x could be 50 degrees or 70 degrees. All we know is that the triangles are isosceles, but we know nothing about the angles.

Statements 1 and 2 together are insufficient. From statement 1 we know x is 70 and that angles BAC and BCA are both 55. But again we know nothing about the placement of point D. Point D could be very close to B in which case y would be close to 70 degrees or point D could be close to the segment AC in which case y would be close to 180 degrees.

I hope that helps! Bhavin

Maybe this is a silly question - I understood this problem but I have one issue with the answer explanation. Why do both angles have to be 55? Isoceles triangles must have at least two equivalent angles, right? Why couldn't the angles have been 70/70/40? Doesn't change the answer from E, but still im curious

Merging similar topics.

As for your question: yes, ABC could be 70-70-40 triangle as well.
_________________

Re: In the figure above, what is the value of x + y ? [#permalink]

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22 Mar 2014, 06:50

Hello,

If we consider 2 statements together then: Why can not these 2 triangles i.e. BAC and DAC are similar by SSS? BA/BC = DA/DC = AC\AC =1 and Hence , <x = <y so then Ans is C.

Re: In the figure above, what is the value of x + y ? [#permalink]

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26 Mar 2014, 01:33

manojISB wrote:

Hello,

If we consider 2 statements together then: Why can not these 2 triangles i.e. BAC and DAC are similar by SSS? BA/BC = DA/DC = AC\AC =1 and Hence , so then Ans is C.

Thanks.

Hi Manoj,

How did you get the ratio of sides as one ?? Can you figure out which sides in the triangle are equal??

Please refer to the solutions above.

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“If you can't fly then run, if you can't run then walk, if you can't walk then crawl, but whatever you do you have to keep moving forward.”

Re: In the figure above, what is the value of x + y ? [#permalink]

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26 Mar 2014, 04:31

(1) x = 70. (2) ABC and ADC are both isosceles triangles.

Per the 2nd statement , I think AB= BC and AD = DC so AB/BC = AD/DC = 1 or AB/AD = BC/DC =1 And ofcourse AC/AC =1 so triangles are similar?? am I wrong here?

(1) x = 70. (2) ABC and ADC are both isosceles triangles.

Per the 2nd statement , I think AB= BC and AD = DC so AB/BC = AD/DC = 1 or AB/AD = BC/DC =1 And ofcourse AC/AC =1 so triangles are similar?? am I wrong here?

Thanks.

From AB/BC = AD/DC = 1 you cannot write AB/AD = BC/DC =1. AB/AD = 1 would mean that AB and AD have the same length which is obviously wrong.
_________________

Re: In the figure above, what is the value of x + y ? [#permalink]

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18 May 2014, 10:14

mikemcgarry wrote:

Bunuel wrote:

Merging similar topics. As for your question: yes, ABC could be 70-70-40 triangle as well.

Bunuel and others, I feel this question has a significant gap --- is the original diagram drawn to scale or not?

Of course, on the GMAT, we know "Figures are drawn as accurately as possible. Exceptions will be noted." I discuss this in a blog post. http://magoosh.com/gmat/2012/gmat-trick ... -possible/ This is a crucial fact for students to keep in mind when interpreting GMAT diagrams.

IF this diagram is purporting to be drawn as accurate as possible, then it's an exceptionally poor diagram. That angle looks nothing like a 70 degree angle. If it's drawn to scale, though, we have to accept that the diagram is somewhat close to symmetrical, and therefore, the 70-70-40 triangle would not be possible.

IF the diagram is not drawn to scale, which I suspect was the intent of the author, that needs to be explicitly stated. Then, the 70-70-40 triangle would be possible. Any bilateral symmetry is out the window if it's not drawn to scale. Here's a scaled diagram of the figure with the 70-70-40 triangle.

Attachment:

isosceles triangles.JPG

I believe, either way, the answer would be (E). Nevertheless, I think this is a crucial issue for students to consider while analyzing the possibilities for a given diagram on the GMAT.

What do others think? Mike

Very important point Mike. I also presumed ( from the diagram in the question that Ab=BC and AD=DC. So how do we tackle such questions in real exam ? Do we not take it scale.

Very important point Mike. I also presumed ( from the diagram in the question that Ab=BC and AD=DC. So how do we tackle such questions in real exam ? Do we not take it scale.

Dear himanshujovi, I'm happy to respond.

On GMAT PS, unless noted, problems are always drawn to scale. Of course, things that look equal may be close by not exactly equal, things that look parallel or perpendicular may be not exactly, etc. For such exact things, you need to rely on what the problem explicitly tells you.

Geometry on the GMAT DS is much trickier, because the diagram could be drawn to scale or not even pretending to be drawn to scale. You need to have a strong visual imagination. One way to approach this --- given a rough DS diagram and the constraints in the question, practice drawing on paper as many variants as possible that are still totally consistent with the explicitly given constraints. It takes some practice to develop your visual intuition, but the more your practice, the more you will be able to see different geometric possibilities.

If, for a specific problem, you would like to see some geometric variants, then post the problem as a new thread in the Magoosh forum: magoosh-324/ and I will post some diagrams of possible altenatives.

Re: In the figure above, what is the value of x + y ? [#permalink]

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18 May 2014, 21:06

Thanks Mike.. As a matter of fact geometry was one of my favourite branch of Mathematics(after Calculus) during my pre engineering days. But then there was a very clear rule - Not to make any assumption at all from any figure unless explicitly mentioned. Looks like GMAT has played with this style of thinking or maybe I need to retune my neural cells [SMILING FACE WITH SMILING EYES]. Thanks anyways for the explanation

Merging similar topics. As for your question: yes, ABC could be 70-70-40 triangle as well.

Bunuel and others, I feel this question has a significant gap --- is the original diagram drawn to scale or not?

Of course, on the GMAT, we know "Figures are drawn as accurately as possible. Exceptions will be noted." I discuss this in a blog post. http://magoosh.com/gmat/2012/gmat-trick ... -possible/ This is a crucial fact for students to keep in mind when interpreting GMAT diagrams.

IF this diagram is purporting to be drawn as accurate as possible, then it's an exceptionally poor diagram. That angle looks nothing like a 70 degree angle. If it's drawn to scale, though, we have to accept that the diagram is somewhat close to symmetrical, and therefore, the 70-70-40 triangle would not be possible.

IF the diagram is not drawn to scale, which I suspect was the intent of the author, that needs to be explicitly stated. Then, the 70-70-40 triangle would be possible. Any bilateral symmetry is out the window if it's not drawn to scale. Here's a scaled diagram of the figure with the 70-70-40 triangle.

Attachment:

isosceles triangles.JPG

I believe, either way, the answer would be (E). Nevertheless, I think this is a crucial issue for students to consider while analyzing the possibilities for a given diagram on the GMAT.

What do others think? Mike

Very important point Mike. I also presumed ( from the diagram in the question that Ab=BC and AD=DC. So how do we tackle such questions in real exam ? Do we not take it scale.

OG13, page 272: A figure accompanying a data sufficiency problem will conform to the information given in the question but will not necessarily conform to the additional information given in statements (1) and (2). Lines shown as straight can be assumed to be straight and lines that appear jagged can also be assumed to be straight. You may assume that the positions of points, angles, regions, and so forth exist in the order shown and that angle measures are greater than zero degrees. All figures lie in a plane unless otherwise indicated.

OG13, page 150: Figures: A figure accompanying a problem solving question is intended to provide information useful in solving the problem. Figures are drawn as accurately as possible. Exceptions will be clearly noted. Lines shown as straight are straight, and lines that appear jagged are also straight. The positions of points, angles, regions, etc., exist in the order shown, and angle measures are greater than zero. All figures lie in a plane unless otherwise indicated.

Re: In the figure above, what is the value of x + y ? [#permalink]

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24 May 2014, 10:33

Bunuel wrote:

In the figure above, what is the value of x + y ?

(1) x = 70. (2) ABC and ADC are both isosceles triangles.

Even when we consider both statements together we don't know the placement of points D and B. For example consider the diagram below:

Attachment:

Triangles2.png

As you can see we can have two different answers for x+y for two different placements of point D.

Answer: E.

Hi Bunuel,

If they had provided us additional information as such: AB=BC, AD=AC and ADC = 1/2 height of ABC. If that's the case, can we use the same inscribed angle rules we use for circle - can we assume that Y would've been 2x, therefore 140 degrees?

Would the answer change if AB & BC weren't equal and AD & AC weren't equal?

(1) x = 70. (2) ABC and ADC are both isosceles triangles.

Even when we consider both statements together we don't know the placement of points D and B. For example consider the diagram below:

Attachment:

Triangles2.png

As you can see we can have two different answers for x+y for two different placements of point D.

Answer: E.

Hi Bunuel,

If they had provided us additional information as such: AB=BC, AD=AC and ADC = 1/2 height of ABC. If that's the case, can we use the same inscribed angle rules we use for circle - can we assume that Y would've been 2x, therefore 140 degrees?

Would the answer change if AB & BC weren't equal and AD & AC weren't equal?

Thanks

Dear russ9, I'm happy to answer this.

If we knew AB=BC and AD=AC, then we would know the triangles were isosceles, which would have added a tremendous amount of useful information to the problem.

Your application of the inscribed angle rule, unfortunately, is 100% incorrect. Among other things, if triangle ABC were inscribed in a circle, the center of that circle would NOT lie on side AC. If ADC = 1/2 height of ABC, then that would NOT mean that AD bisected angle BAC or that CD bisected angle BCA. If we had been told that those two segments were angle bisectors, then yes, it would have been true that y = 140.

Here's a real paradox: if we had been told the three pieces of information you suggested, those would uniquely determine triangle ADC, such that all of its angles would be mathematically determined. To find the measure of y would involves some sophisticated trigonometry and a calculator --- finding the numerical value of y would be well beyond anything the GMAT could expect you to do, but technically, y would be mathematically determined and one could find it, which means that information would be sufficient. In practice, the GMAT doesn't put you in that position --- it doesn't expect you to find as sufficient something that is uniquely determined mathematically but incalculable using ordinary GMAT math. The GMAT will only put you in the position of declaring something sufficient if you realistically could find it with ordinary GMAT math.

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