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If points A and C both lie on the circle with center B and the measure

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Bunuel
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If points A and C both lie on the circle with center B and the measure [#permalink] New post   25 Jul 2018, 00:54
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If points A and C both lie on the circle with center B and the measurement of angle ABC is not a multiple of 30, what is the ratio of the area of the circle centered at point B to the area of triangle ABC?


A) 2π

B) \(\frac{2π(AB)^2}{(BC)^2}\)

C) 4π

D) \(\frac{π(BC)^2}{.5(BC)(AB)}\)

E) None of the Above

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chetan2u
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Re: If points A and C both lie on the circle with center B and the measure [#permalink] New post   25 Jul 2018, 01:05
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Bunuel wrote:

If points A and C both lie on the circle with center B and the measurement of angle ABC is not a multiple of 30, what is the ratio of the area of the circle centered at point B to the area of triangle ABC?


A) 2π

B) \(\frac{2π(AB)^2}{(BC)^2}\)

C) 4π

D) \(\frac{π(BC)^2}{.5(BC)(AB)}\)

E) None of the Above

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Area of \(\triangle{ABC}\) is dependent on \(\angle{ABC}\) and since we do not know the exact angle, we cannot find this area..
so we cannot find the ratio of the area involving the area of triangle ABC
so E
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Re: If points A and C both lie on the circle with center B and the measure [#permalink] New post   25 Jul 2018, 13:58
If the triangle ABC is an isosceles triangle, where the radius of the circle is equal to x, wouldn’t the answer be 2pi?

Area of the circle = (x)(x)pi
Area of triangle abc = (1/2) base*height = (1/2)(x)(x)

So the ratio would simplify to 2pi. Right?

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If points A and C both lie on the circle with center B and the measure [#permalink] New post   25 Jul 2018, 14:19
Probably, easiest way to solve is to use trigonometry.

Area of circle= πr^2= π(AB)^2

Area of triangle= 1/2*AB*BC*SinABC= 1/2*(AB)^2*SinB

Ratio of Area of Circle: Triangle= π(AB)^2 / 0.5*(AB)^2*SinB = 2π/SinB

Now, as angle B cannot be 90 (multiple of 30), Sin B cannot be equal to 1. Thus, Ratio cannot be 2π. So, choices A, B and D are out as each one can be simplified to 2π.

Similarly, angle B cannot be 30, so Sin B cannot be equal to 1/2. Hence, Ratio cannot be 4π. So, Choice C is out.

We are left with Ans E.

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Re: If points A and C both lie on the circle with center B and the measure [#permalink] New post   30 Jul 2018, 12:49
I see the options A,B and D are different representations of the same. But why could the answer not be any one of these? Also is it really necessary to know the angle to find the area of the triangle? isnt the base and height enough? please explain. Thank you!
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Re: If points A and C both lie on the circle with center B and the measure [#permalink] New post   30 Jul 2018, 12:56
Steven1604 wrote:
I see the options A,B and D are different representations of the same. But why could the answer not be any one of these? Also is it really necessary to know the angle to find the area of the triangle? isn't the base and height enough? please explain. Thank you!


Steven1604,

The area of a triangle are dependent on base and height. But for a triangle with one vertex at the circle and two points on the circle, both height & base depend on the angle it subtends at the center. Instead of going into the math - just imagine a triangle with base which is very small such that the triangle is barely visible ( here the angle at the center would be close to zero ) vs. a triangle with angle > 90 at center would be much larger..

Without thinking about the base and the height here - we know for sure a smaller triangle ill have a smaller area... so the ratio of areas cannot be found!

Hope this makes sense.

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Re: If points A and C both lie on the circle with center B and the measure [#permalink] New post   26 Jun 2021, 16:16
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Re: If points A and C both lie on the circle with center B and the measure [#permalink]
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