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kcui
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Post subject: Inscribed Triangle, Data Suff - Could someone explain?  Posted: Tue May 20, 2008 11:53 am |
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The data sufficiency question on the CAT is as follows:
For the triangle shown above, where A, B and C are all points on a circle, and line segment AB has length 18, what is the area of triangle ABC?
(1) Angle ABC measures 30°.
(2) The circumference of the circle is 18*PI.
In order to find the area of the triangle, we need to find the lengths of a base and its associated height. Our strategy will be to prove that ABC is a right triangle, so that CB will be the base and AC will be its associated height.
(1) INSUFFICIENT: We now know one of the angles of triangle ABC, but this does not provide sufficient information to solve for the missing side lengths.
(2) INSUFFICIENT: Statement (2) says that the circumference of the circle is 18. Since the circumference of a circle equals times the diameter, the diameter of the circle is 18. Therefore AB is a diameter. However, point C is still free to "slide" around the circumference of the circle giving different areas for the triangle, so this is still insufficient to solve for the area of the triangle.
(1) AND (2) SUFFICIENT: Note that inscribed triangles with one side on the diameter of the circle must be right triangles. Because the length of the diameter indicated by Statement (2) indicates that segment AB equals the diameter, triangle ABC must be a right triangle. Now, given Statement (1) we recognize that this is a 30-60-90 degree triangle. Such triangles always have side length ratios of
1::2
Given a hypotenuse of 18, the other two segments AC and CB must equal 9 and 9 respectively. This gives us the base and height lengths needed to calculate the area of the triangle, so this is sufficient to solve the problem.
The correct answer is C. (Both Together are Sufficient)
My question is, isn't the first point sufficient? The explaination claims the info in point 1 is not sufficient to find the missing lengths.
If I remember correctly, given an angle and a hypotenuse, isn't is possible to determine the other sides via trigonometry? If one angle is 30 degrees, and the hypotenuse 18, then the lengths of other two sides are 18 SIN (30) and 18 COS(30).
Or am I being stupid and missing something basic?
Thanks
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Abhimanyu
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Post subject: Posted: Wed May 21, 2008 11:02 am |
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Hey Kcui....
I need two clarifications from you on this Question
1.) Is the Line segment AB is the Diameter of the Circle?
2.) Is the Triangle Inscribed in the Semi Circle with hypotenuse as its Diameter?
If Answer to both these questions is yes, Then Answer to your question is A
Statement 1 will create a scenario of 30 60 90 Triangle which has the side ratio of 1 Square Root (3) and 2, and here we know one side which is 18 and thus can determine other side and hence the area of the triangle.
Statement 2 doesn’t provides any clue to the Angles of the of the right angle triangle hence we cannot determine the length of other sides......
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RonPurewal
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Post subject: Re: Inscribed Triangle, Data Suff - Could someone explain? Posted: Wed May 28, 2008 7:35 am |
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kcui wrote: If I remember correctly, given an angle and a hypotenuse, isn't is possible to determine the other sides via trigonometry? If one angle is 30 degrees, and the hypotenuse 18, then the lengths of other two sides are 18 SIN (30) and 18 COS(30).
Or am I being stupid and missing something basic?
Thanks
i wouldn't say stupid; it's a very common oversight. but you can't use the above trig functions unless you have ascertained that you're dealing with a RIGHT triangle. you don't have enough evidence to say that for sure unless you have statement 2, which ensures that AB is a diameter (because only diameters can have length = circumference divided by pi)
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sridefies
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Post subject: Re: Inscribed Triangle, Data Suff - Could someone explain? Posted: Thu Dec 24, 2009 9:45 am |
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Isnt there property of semicircles that "Angles within a semicircle is always 90"
In that case,,isnt option B sufficient?
Pls help..
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Ben Ku
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Post subject: Re: Inscribed Triangle, Data Suff - Could someone explain? Posted: Tue Feb 16, 2010 8:44 pm |
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sridefies wrote: Isnt there property of semicircles that "Angles within a semicircle is always 90"
In that case,,isnt option B sufficient?
Pls help.. B helps us know that ABC is a right triangle. However, the question is asking for the area. Just knowing that AB is the diameter does not tell us the height of the right triangle. Hope that makes sense.
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Ben Ku
Instructor
ManhattanGMAT
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guy.b
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Post subject: Re: Inscribed Triangle, Data Suff - Could someone explain? Posted: Sat Jul 31, 2010 3:12 am |
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Hi,
when I read the question and saw the image, I assumed that angle ACB is right angle because it was not written that "figure not drawn to scale". What should I do in the GMAT axam? can I assume right angle? Why the comment "figure not drawn to scale" is not there?
and one more question- how can I insert the image?
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mschwrtz
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Post subject: Re: Inscribed Triangle, Data Suff - Could someone explain? Posted: Sun Aug 22, 2010 1:09 pm |
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nelvin898
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Post subject: Re: Inscribed Triangle, Data Suff - Could someone explain? Posted: Wed Aug 31, 2011 2:17 pm |
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Ben Ku wrote: sridefies wrote: Isnt there property of semicircles that "Angles within a semicircle is always 90"
In that case,,isnt option B sufficient?
Pls help.. B helps us know that ABC is a right triangle. However, the question is asking for the area. Just knowing that AB is the diameter does not tell us the height of the right triangle. Hope that makes sense. Hi Ben, Can you correct me if I am wrong, the figure looks like one side of the triangle is the diameter of the circle, BUT IT DOESN'T SAY SO. It just indicates that the length of the side is 18. But it looks like it. So is it safe to assume, that if it were so, it would be clearly stated? I just don't want to repeat the same mistake of assuming it is when it's not, or mistaking it as a NO when it is.
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jnelson0612
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Post subject: Re: Inscribed Triangle, Data Suff - Could someone explain? Posted: Sat Oct 01, 2011 4:03 pm |
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Am I alone in being unable to see the figure? When I click Michael's link I don't see a diagram.
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Jamie Nelson
ManhattanGMAT Instructor
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specialxknc22
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Post subject: Re: Inscribed Triangle, Data Suff - Could someone explain? Posted: Tue Feb 28, 2012 7:01 am |
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I ran across this problem too and assumed that the line was the diameter. Is this an incorrect assumption? I didn't say, 'not drawn to scale.'
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jnelson0612
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Post subject: Re: Inscribed Triangle, Data Suff - Could someone explain? Posted: Sat Mar 03, 2012 11:31 pm |
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specialxknc22 wrote: I ran across this problem too and assumed that the line was the diameter. Is this an incorrect assumption? I didn't say, 'not drawn to scale.' It's hard for me to say much about this since I still can't see the figure, but let me just say as a general principle not to assume something like that. If you can derive that information using geometric theorems it's fine, but just eyeballing it is not adequate proof that the line is the diameter.
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Jamie Nelson
ManhattanGMAT Instructor
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