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

Question

https://gmatclub.com/forum/download/file.php?id=23448 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 . Untitled.png

dentobizz · Accepted Answer

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.

enigma123 · Answer

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?

dentobizz · Answer

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

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.

CA88 · Answer

Answer is C IMO.

(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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honchos · Answer

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?

Bunuel · Answer

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?

pacifist85 · Answer

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?

EMPOWERgmatRichC · Answer

Hi pacifist85,

While "adding in" a radius can be a useful 'step' in dealing with a multiple-shape Geometry question, it won't actually help us here.

With the information in Fact 2, we know that we have a right triangle with a hypotenuse of 18, but we don't know ANYTHING else.

With that info, we have A^2 + B^2 = 18^2, but we don't know the actual values of A and B (which we need to figure out the area). If we had info about the other angles, then we COULD figure those sides out though (since we'd have a relationship among the 3 sides based on the angles).

Adding in a radius won't help us figure out any of the other angles (in the big triangle or in either of the smaller triangles).

To figure out the area of the triangle, we need a "base" and a "height." We COULD set the 18 as the base, but we have no way of determining the height without additional information (at least one more of the sides or one of the non-90 degree angles). As such, Fact 2 is INSUFFICIENT.

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

pacifist85 · Answer

Hi Rich,

I understand eveything you said. Actually, I wasn't assuming any of this. I added the image to illustrate what I thought.

So, after drawing it, I realised the mistake in my line of reasoning. The problem is that I cannot be sure which one the hypotenuse is, in any of the 2 smaller triangles, so using the pythagorean I can find the third side. Right?

Because all I know is that AC + CB >18 and that 18> CB - AC. But even with this information, I cannot figure out which one of the sides has the bigger or smaller length.

I was focusing on the lower triangle, because it seems that CB is greater than the radius, but I cannot be sure. So, "it seems" is not enough.

EMPOWERgmatRichC · Answer

Hi pacifist85,

A hypoteneuse ONLY exists in right triangles, so neither of the two small triangles actually has a hypoteneuse.

When dealing with Geometry questions on the GMAT, it's often really helpful to draw the physical shapes (as opposed to just staring at words or sticking to just the math formulas involved). You'll be far more likely to make 'connections' and the proper deductions when you can "see" the shapes and your work, so I encourage you to continue doing what you did here - draw the pictures and get the points.

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

anairamitch1804 · Answer

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: sqrt3 :2

Given a hypotenuse of 18, the other two segments AC and CB must equal 9 and 9sqrt3 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.

anuj11 · Answer

Why is B not enough ? If AB is the diameter then triangle ABC has to be 30-60-90 triangle and if that's the case then we can calculate the area right ?

I had read somewhere that if the Hypotenuse is confirmed as the diameter then the triangle is always a right angle triangle.

EMPOWERgmatRichC · Answer

Hi anuj11,

With the information in Fact 2, we know that we have a right triangle with a hypotenuse of 18, but we don't know ANYTHING else.

With that info, we have A^2 + B^2 = 18^2, but we don't know the actual values of A and B (which we need to figure out the area). If we had info about the other ANGLES, then we COULD figure those sides out though (since we'd have a relationship among the 3 sides based on the angles). However, we don't know whether the triangle is a 30/60/90 or some other right triangle... and those different options would have different areas. As such, Fact 2 is INSUFFICIENT.

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

AdityaSurana94 · Answer

Can any one please explain why B alone is not sufficient.? 
Is it because angle A and B could either of measures (i.e. 45,45 or 30,60).?

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Bunuel · Answer

The second statement is not sufficient because even though we know that ABC must be a right triangle we don't know measures of its remaining angles. It's not necessary ABC to be either 30-60-90 or 45-45-90, it can be any right triangle, say 0.5-89.5-90 or say 17-73-90, ...

Hope it helps.

Hope it helps.

RMD007 · Answer

Bunuel  If it were given in the question that angle ACB = 90, then can we conclude that AB is the diameter of the circle?

Bunuel · Answer

Yes.

A right triangle inscribed in a circle must have its hypotenuse as the diameter of the circle. The reverse is also true: if the diameter of the circle is also the triangle’s hypotenuse, then that triangle is a right triangle.

GMATYoda · Answer

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:\sqrt{3}: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.

mansi1997 · Answer

Isn't this a property : A right triangle inscribed in a circle must have its hypotenuse as the diameter of the circle.
If we use this, we get Statement 1 sufficient?

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

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

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