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If points A, B, and C lie on a circle of radius 1, what is [#permalink]
19 Mar 2011, 12:20

00:00

A

B

C

D

E

Difficulty:

5% (low)

Question Stats:

46% (02:44) correct
53% (00:38) wrong based on 13 sessions

Quote:

If points A, B, and C lie on a circle of radius 1, what is the area of triangle ABC?

1. AB^2 =AC^2+BC^2 2. Angle CAB equals 30 degrees

The previous answers in this forum tended for C as the correct answer. I've marked B not C and let me explain why

statement (1) suggests that there's a right triangle, BUT the angle sides might be different and the area of triangle might vary with these angle mesaures. E.g. when angles follow 45-45-90 the area of triangle would be 1, while with 30-60-90 the area of triangle is Sqrt(3)/2 Not Sufficient;

statement (2) Very interesting statement offering the inscribed angle measurement. If we find the angle CAB intercepted at the center, we get (30`)*2 OR 60`. Additionally, with the centrally intercepted angle we have the isosceles triangle with the base angles 60` which convert into the equilateral triangle, since all angles are 60` (BC=OC=OB). SO, side BC is equal to radius 1.

If we continue the line BO from the point O up-to the point D we receive height DC for the side BC of triangle ABC. Now we need to calculate the height which is easy by knowing triangle BCD is a right triangle and angle CBD=60`. So, DC is Sqrt(3). The area of triangle ABC using all these properties ---> base (BC)*height (CD)/2 = 1*Sqrt(3)/2, Sufficient as we can answer the questions area of triangle ABC=Sqrt(3)/2 therefore answer B.

Re: Area of Triangle inside a Circle [#permalink]
19 Mar 2011, 18:55

Expert's post

zaur2010 wrote:

Quote:

If points A, B, and C lie on a circle of radius 1, what is the area of triangle ABC?

1. AB^2 =AC^2+BC^2 2. Angle CAB equals 30 degrees

The previous answers in this forum tended for C as the correct answer. I've marked B not C and let me explain why

statement (1) suggests that there's a right triangle, BUT the angle sides might be different and the area of triangle might vary with these angle mesaures. E.g. when angles follow 45-45-90 the area of triangle would be 1, while with 30-60-90 the area of triangle is Sqrt(3)/2 Not Sufficient;

statement (2) Very interesting statement offering the inscribed angle measurement. If we find the angle CAB intercepted at the center, we get (30`)*2 OR 60`. Additionally, with the centrally intercepted angle we have the isosceles triangle with the base angles 60` which convert into the equilateral triangle, since all angles are 60` (BC=OC=OB). SO, side BC is equal to radius 1.

If we continue the line BO from the point O up-to the point D we receive height DC for the side BC of triangle ABC. Now we need to calculate the height which is easy by knowing triangle BCD is a right triangle and angle CBD=60`. So, DC is Sqrt(3). The area of triangle ABC using all these properties ---> base (BC)*height (CD)/2 = 1*Sqrt(3)/2, Sufficient as we can answer the questions area of triangle ABC=Sqrt(3)/2 therefore answer B.

First of all, I think it's a great effort. It is always refreshing when people try to analyze from different perspectives. There was one error though... Look at the diagram below and figure out which of the following colorful altitudes could help you find the area of the triangle? They are all perpendicular to their respective bases.

Attachment:

Ques2.jpg [ 7.77 KiB | Viewed 2723 times ]

I think you will agree that the purple line cannot be used as an altitude to find the area of this triangle... I hope this helps you in identifying your mistake.
_________________

Re: Area of Triangle inside a Circle [#permalink]
19 Mar 2011, 19:21

Except for the rt angled triangle the geometry cannot be defined with two parameters. First assume S1 right angled -one parameter (hyp is known). The other parameter is S2 (one angle is known)

Re: Area of Triangle inside a Circle [#permalink]
29 Mar 2011, 11:18

zaur2010 wrote:

If points A, B, and C lie on a circle of radius 1, what is the area of triangle ABC?

1. AB^2 =AC^2+BC^2 2. Angle CAB equals 30 degrees

my take is A as by having 1 AB^2 =AC^2+BC^2 we are clear that it will be a right angle triangle with right angle at C and then AB will be the diameter = 2 and then we know all the angles hence can find out the area

while with statement 2. Angle CAB equals 30 degrees there are various possibilities of triangles with different hieghts and diffferent bases

please clarify
_________________

WarLocK _____________________________________________________________________________ The War is oNNNNNNNNNNNNN for 720+ see my Test exp here http://gmatclub.com/forum/my-test-experience-111610.html do not hesitate me giving kudos if you like my post.

Re: If points A, B, and C lie on a circle of radius 1, what is [#permalink]
27 Dec 2013, 15:02

zaur2010 wrote:

Quote:

If points A, B, and C lie on a circle of radius 1, what is the area of triangle ABC?

1. AB^2 =AC^2+BC^2 2. Angle CAB equals 30 degrees

The previous answers in this forum tended for C as the correct answer. I've marked B not C and let me explain why

statement (1) suggests that there's a right triangle, BUT the angle sides might be different and the area of triangle might vary with these angle mesaures. E.g. when angles follow 45-45-90 the area of triangle would be 1, while with 30-60-90 the area of triangle is Sqrt(3)/2 Not Sufficient;

statement (2) Very interesting statement offering the inscribed angle measurement. If we find the angle CAB intercepted at the center, we get (30`)*2 OR 60`. Additionally, with the centrally intercepted angle we have the isosceles triangle with the base angles 60` which convert into the equilateral triangle, since all angles are 60` (BC=OC=OB). SO, side BC is equal to radius 1.

If we continue the line BO from the point O up-to the point D we receive height DC for the side BC of triangle ABC. Now we need to calculate the height which is easy by knowing triangle BCD is a right triangle and angle CBD=60`. So, DC is Sqrt(3). The area of triangle ABC using all these properties ---> base (BC)*height (CD)/2 = 1*Sqrt(3)/2, Sufficient as we can answer the questions area of triangle ABC=Sqrt(3)/2 therefore answer B.

I think I may have made a mistake

See, from Statement 1 we have that ABC is a right triangle, since it is inscribed in the circle then of course hypothenuse = diameter = 2

So then how can we find the area. Well can't we extend a height perpendicular to the diameter which will in fact be the radius = 1 to find it. With base and height we could have the area

Would anybody be so kind to explain why this reasoning is wrong?

Re: If points A, B, and C lie on a circle of radius 1, what is [#permalink]
27 Dec 2013, 18:47

Jlgdr,Warlock, I think you are considering the triangle as isosceles triangle with diameter AB. But the point C can be very near to say Point A or B and still be a right angle and statement 1 would be true. But the height ( perpendicular on AB from C ) would change and thus the area.