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If points A, B, and C lie on a circle of radius 1, what is [#permalink]
 19 Mar 2011, 13:20
Question Stats:
25% (05:09) correct
75% (00:36) wrong based on 8 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.
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Re: Area of Triangle inside a Circle [#permalink]
19 Mar 2011, 13:29
The answer is C. Take a minute and think why the height of triangle ABC is CD. The height of triangle ABC should AE where E is the point extended the line BC from C.
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Re: Area of Triangle inside a Circle [#permalink]
19 Mar 2011, 14:16
Ans is C. Possible inscribed triangles with 30^{\circ} angle. All these triangles have different areas.
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Re: Area of Triangle inside a Circle [#permalink]
19 Mar 2011, 19:55
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 2196 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.
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Re: Area of Triangle inside a Circle [#permalink]
19 Mar 2011, 20:21
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) Posted from my mobile device 
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Re: Area of Triangle inside a Circle [#permalink]
29 Mar 2011, 12:18
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 
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Re: Area of Triangle inside a Circle [#permalink]
29 Mar 2011, 18:17
@zaur2010, please extend line BC upwards and draw a perpendicular line from A dropping on that line, that will be the height of the triangle. I don't know how to draw Geometry figures online, else would have done so.
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Re: Area of Triangle inside a Circle
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29 Mar 2011, 18:17
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