Inscribed Triangle, Data Suff - Could someone explain?

[kcui]

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

[Abhimanyu]

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......

[RonPurewal]

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)

[sridefies]

Isnt there property of semicircles that "Angles within a semicircle is always 90"
In that case,,isnt option B sufficient?
Pls help..

[Ben Ku]

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.

[guy.b]

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?

[mschwrtz]

see

figure-t10673.html

[nelvin898]

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.

[jnelson0612]

Am I alone in being unable to see the figure? When I click Michael's link I don't see a diagram.

[specialxknc22]

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.'

[jnelson0612]

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.

[mbakliye]

Hi,

Can we use the theorem(sorry I don't remember the name of the theorem) that states : the angle at the centre of a circle is twice the angle at the circumference if both angles stand on the same arc.
If yes , then is the option A not sufficient ?

Angle AOC = 60 (By the above stated theorem, Angle AOC = 2* Angle ABC)
Angle CAB = 60 (as the angle COB = 180 - Angle AOC = 120, By the above stated theorem, Angle COB = 2* Angle CAB)
Angle ACO = 60 (In triangle ACO, sum of the angles in a triangle = 180)
Angle BCO = 30 (In triangle BCO, sum of the angles in a triangle = 180)

Then it is 30-60-90 triangle , and we can calculate the area.

Regards,
Shikhar Hasija

[mbakliye]

Apologies, the answer should be C.

I incorrectly considered , o - the centre of the circle on line AB.

Regards,
Shikhar Hasija

[jnelson0612]

Apologies, the answer should be C.

I incorrectly considered , o - the centre of the circle on line AB.

Regards,
Shikhar Hasija

Okay, good. :-)

[rustom.hakimiyan]

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,


1) Statement 1 says that ABC is 30 degree. Doesn't that imply that we have a 30,60,90? What makes it insufficient is that we don't know if 18 is the diameter, and therefore, we don't know if C or A is 90degree. Is that correct?

So in a nutshell, we can assume that it's a 30,60,90 b/c of the inscribed plus an angle given, but we just don't know where the 60 and 90 sit. Right?

2) Statement 2 states that 18 is the diameter.Doesn't that mean that C HAS to be 90 degrees? So regardless of what A is and what B is, don't we still get the base/height combo?

Thanks