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# For the triangle shown, where A, B and C are all points on

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For the triangle shown, where A, B and C are all points on [#permalink]

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05 Feb 2012, 17:33
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For the triangle shown, 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 $$18\pi$$.
[Reveal] Spoiler: OA

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Re: Area of Triangle ABC [#permalink]

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05 Feb 2012, 17:46
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Statement 1 is insufficient because--We are NOT given that the line segment AB=18 is the diameter of the circle,Hence we CANNOT assume that angle ACB is 90 (ONLY if the hypotenuse is the diameter then the angle opposite to it can be 90) and thus the 30-60-90 formula cannot be applied here using statement 1 alone.

by using statement 2 we can find the radius =9 ( C=2*pi*r=18 pi; r=9) , diameter = 18 , making AB the diameter (AB is given as 18) and thus angle ACB can be taken as 90 and using the info in statement 1 we can apply the 30-60-90 formula; get values of AC and BC and solve for the area of triangle ACB ( 0.5* AC *BC) .Answer C.

Hope it is clear now.
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Re: Area of Triangle ABC [#permalink]

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05 Feb 2012, 18:12
But again the same question - in the question no where it says that line AB passes through the centre of the circle so how can we take it as a diameter?
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Re: Area of Triangle ABC [#permalink]

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05 Feb 2012, 18:26
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enigma123 wrote:
But again the same question - in the question no where it says that line AB passes through the centre of the circle so how can we take it as a diameter?

The diameter is the longest chord/line segment of the circle and all lines passing through the center of the circle and touching the circumference will be the diameters-- that is, they will have the same length,
hence in this question too when we find that AB=18 then we NOW KNOW that it IS the diameter of the circle [NO other line can have a length greater than 18 , and all line less than 18 will not be diameters]

So we are NOT assuming AB to be the diameter, rather when we calculate that diameter =18 (from st.2 )and we are already given AB=18 (question stem) ,then we just put these 2 pieces of information together, to get AB=18=Diameter. Hope this helps.
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09 Sep 2012, 21:58
Hi Guys,

Could you please clarify my doubt for the problem attached as a pic file?

My Doubt : Here the diagram itself shows AB is the diameter of the circle. And statement A confirms that it is a 30-60-90 triangle by mentioning about angle. Why can’t either statement A alone be sufficient.

Thanks,

Harikris
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10 Sep 2012, 02:36
harikris wrote:
Hi Guys,

Could you please clarify my doubt for the problem attached as a pic file?

My Doubt : Here the diagram itself shows AB is the diameter of the circle. And statement A confirms that it is a 30-60-90 triangle by mentioning about angle. Why can’t either statement A alone be sufficient.

Thanks,

Harikris

Just by seeing the diagram one cannot predict whether the line is diameter or not. untill and unless sthing is written either on the diagram or on the text one cannot say anything.
So for me 'C' will be the right choice.
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10 Sep 2012, 02:54
jitgoel wrote:
harikris wrote:
Hi Guys,

Could you please clarify my doubt for the problem attached as a pic file?

My Doubt : Here the diagram itself shows AB is the diameter of the circle. And statement A confirms that it is a 30-60-90 triangle by mentioning about angle. Why can’t either statement A alone be sufficient.

Thanks,

Harikris

Just by seeing the diagram one cannot predict whether the line is diameter or not. untill and unless sthing is written either on the diagram or on the text one cannot say anything.
So for me 'C' will be the right choice.

If the question stem never explicitly tells you that AB is the diameter of the circle, you can't assume that it is the diameter based on the diagram.

(1) Since you can't assume that AB is the diameter (and that ACB is 90 degrees), knowing that ABC is 30 degrees is insufficient.

(2) This statement establishes that ACB is 90 degrees, but that's all. Insufficient.

C. Need to establish that both ACB is 90 degrees and ABC is 30 degrees.
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10 Sep 2012, 03:32
harikris wrote:
Hi Guys,

Could you please clarify my doubt for the problem attached as a pic file?

My Doubt : Here the diagram itself shows AB is the diameter of the circle. And statement A confirms that it is a 30-60-90 triangle by mentioning about angle. Why can’t either statement A alone be sufficient.

Thanks,

Harikris

Lets see one statement at a time.
Statement 1 - It just says that Angle B is 30 degrees but does not say whether AB is the diameter or not. it also doesn't say anything about angle A or C. Thus insufficient.

Statement 2 - circumference is 18pie i.e diameter = 18 which means AB is the diameter & angle C = 90. But it does not state anything about angle A or B. Thus insufficient.

Statement 1 & 2 - AB = diameter, angle B= 30 , C=90 & A=60. Using 30-60-90 we can calculate the area of the triangle.

Hope it helps
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10 Sep 2012, 08:57
The diagram is an illusion. AB really seems to be the diameter. One can easily ignore statement 2 and assume AB as diameter.

But careful thinking makes it clear that AB cannot be assumed as diameter unless we consider statement 2.
Then we can proceed to the solution with 60-90-30 degree angle.

As per me, we should be careful about such pitfall.
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For the triangle shown, where A, B and C are all points on [#permalink]

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12 Oct 2012, 13:54
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.

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Re: For the triangle shown above, where A, B and C are all [#permalink]

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12 Oct 2012, 16:08
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(1) ABC 60 deg, other angles of triangle could be anything. Could be equilateral triangle, could be 30-60-90. Insuff.

(2) circ is 18 pi tells us that AB is diameter or circle. Without any angles, triangle could be 30-60-90 or 45-45-90. Insuff.

Together, triangle is 30-60-90 with hypotenuse 18. Sufficient

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Re: For the triangle shown above, where A, B and C are all [#permalink]

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12 Oct 2012, 16:22
@shikhar

That would work if you are given O (I'm guessing you're referring to the center of the circle) is on AB. If O does not lie on AB you wouldn't be able to establish angle COB.

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Re: For the triangle shown above, where A, B and C are all [#permalink]

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13 Oct 2012, 01:06
@HImba88 : Yes , you are correct

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Re: For the triangle shown, where A, B and C are all points on [#permalink]

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17 Jun 2014, 10:46
I think C is not correct.

A alone is sufficient.

Hypotenuse is given = 18

we know that diameter makes a right angle over the circumference C = 90degree

so from A alone we conclude:

it is 30-60-90 triangle One side is known other two sides can be calculated, Hence area of triangle ABC can be calculated, Do you agree Bunuel?
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Re: For the triangle shown, where A, B and C are all points on [#permalink]

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18 Jun 2014, 09:15
honchos wrote:
I think C is not correct.

A alone is sufficient.

Hypotenuse is given = 18

we know that diameter makes a right angle over the circumference C = 90degree

so from A alone we conclude:

it is 30-60-90 triangle One side is known other two sides can be calculated, Hence area of triangle ABC can be calculated, Do you agree Bunuel?

Let me ask you a question: how do you know that AB is the diameter there? How do you know that the triangle is right angled?
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Re: For the triangle shown, where A, B and C are all points on [#permalink]

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03 Aug 2014, 17:59
Bunuel wrote:
honchos wrote:
I think C is not correct.

A alone is sufficient.

Hypotenuse is given = 18

we know that diameter makes a right angle over the circumference C = 90degree

so from A alone we conclude:

it is 30-60-90 triangle One side is known other two sides can be calculated, Hence area of triangle ABC can be calculated, Do you agree Bunuel?

Let me ask you a question: how do you know that AB is the diameter there? How do you know that the triangle is right angled?

Not the original poster, but I have a similar questions:

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?

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Re: For the triangle shown, where A, B and C are all points on [#permalink]

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03 Aug 2014, 21:06
@Russ9

"Not the original poster, but I have a similar questions:

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?

I don't know how you get that, there is no rule that any triangle inscribed inside a circle should be a right triangle. if ABC is 30, why should the other two be 90,60? why not 100,50 or 80,70? how do you assert, that it implies that its a 30,60,90 triangle? Yes, if 18 is the diameter, then its a right triangle, then AB would be a right triangle.. otherwise it could be anything.

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?

yes in this case C is 90. and AB=18 is the diameter, but how will you calculate area with that? base/height combo? what is base and height here? how would you determine that? Do all right angles triangle with the same hypotenuse have the same area? No. We need at least one angle or at least the length of one of the sides to calculate the area.
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Re: For the triangle shown, where A, B and C are all points on [#permalink]

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12 Oct 2014, 09:07
After reading through the explanations above, I'm still confused as to why we can't assume outright that AB is the diameter since the stem tells us that points ABC are all on the circle. Isn't there a rule that when a triangle is inscribed in a circle it is automatically a right triangle?
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Re: For the triangle shown, where A, B and C are all points on [#permalink]

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12 Oct 2014, 12:05
macey15 wrote:
After reading through the explanations above, I'm still confused as to why we can't assume outright that AB is the diameter since the stem tells us that points ABC are all on the circle. Isn't there a rule that when a triangle is inscribed in a circle it is automatically a right triangle?

No, that's totally wrong. Inscribed triangle is right only if a side of a triangle coincides with the diameter of a circle.
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Re: For the triangle shown, where A, B and C are all points on [#permalink]

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13 Feb 2015, 13:48
Well, I did it wrong assuming too much myself and I understand my mistakes. All apart from one...

So, statement 2 says that he circumference of the circle is 18π. With this we can find r.

So, 18π = 2πr --> r = 9. Since r = 9, then AB = 18 is the diameter. Why cannot we draw a line from the center of the circle to angle C, which would mean it would be one radius of the circle? Then, we divide ABC into 2 smaller trianlges, two sides of which equal to 9. We can then use the Pythagorean to find AB and CB. So, then we can find the area...

What am I missing?
Re: For the triangle shown, where A, B and C are all points on   [#permalink] 13 Feb 2015, 13:48

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