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# Is the area of the triangle drawn in circle A greater than π?

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Math Expert
Joined: 02 Sep 2009
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Is the area of the triangle drawn in circle A greater than π?  [#permalink]

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13 Sep 2018, 04:04
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Difficulty:

55% (hard)

Question Stats:

54% (01:33) correct 46% (01:25) wrong based on 69 sessions

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Is the area of the triangle drawn in circle A greater than π? (Assume A to be the center of the circle)

(1) x = 36
(2) The radius of the circle is 1.

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Re: Is the area of the triangle drawn in circle A greater than π?  [#permalink]

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15 Sep 2018, 02:25
Is the area of the triangle drawn in circle A greater than π? (Assume A to be the center of the circle)

(1) x = 36°
you cant calculate area of a triangle with just the angle, each side could be any unit with angle 36....

(2) The radius of the circle is 1.
Area of triangle formula = (1/2)b*h
base = height = radius = 1
area of the triangle = 0.5
Sufficient

Do let me know if my reasoning is right...... thanks
Manager
Joined: 21 Jun 2017
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Re: Is the area of the triangle drawn in circle A greater than π?  [#permalink]

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27 Sep 2018, 08:35
AvidDreamer09 wrote:
Is the area of the triangle drawn in circle A greater than π? (Assume A to be the center of the circle)

(1) x = 36°
you cant calculate area of a triangle with just the angle, each side could be any unit with angle 36....

(2) The radius of the circle is 1.
Area of triangle formula = (1/2)b*h
base = height = radius = 1
area of the triangle = 0.5
Sufficient

Do let me know if my reasoning is right...... thanks

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Re: Is the area of the triangle drawn in circle A greater than π?  [#permalink]

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27 Sep 2018, 08:37
1 Insuff
2 Area of entire circle is pi*1^2= pi
Now if the ares of entire circle is pi, the area of triangle in all cases considered will be less than the area of circle (= pi)
So, the area of traingle must be less than Pi
Sufficient.
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Re: Is the area of the triangle drawn in circle A greater than π?  [#permalink]

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27 Sep 2018, 09:04
2
1. Neither radius nor any side measure is given in the question. Hence angle alone is not sufficient

2. If the area of the entire circle= pi. 1^2 = pi. Therefore area of the triangle will be less than pi for sure . Hence B is sufficient

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Joined: 19 Apr 2017
Posts: 45
Re: Is the area of the triangle drawn in circle A greater than π?  [#permalink]

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27 Sep 2018, 09:06
Kritika2008poddar wrote:
1. Neither radius nor any side measure is given in the question. Hence angle alone is not sufficient

2. If the area of the entire circle= pi. 1^2 = pi. Therefore area of the triangle will be less than pi for sure . Hence B is sufficient

Posted from my mobile device

oooh Thanks... that makes more sense... i dunno what i was doing
GMATH Teacher
Status: GMATH founder
Joined: 12 Oct 2010
Posts: 538
Re: Is the area of the triangle drawn in circle A greater than π?  [#permalink]

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27 Sep 2018, 15:11
1
Bunuel wrote:

Is the area of the triangle drawn in the circle (with center A) greater than $$\pi$$ ?

(1) x = 36 degrees
(2) The radius of the circle is 1.

$${S_\Delta }\,\,\mathop > \limits^? \,\,\pi$$

(1) In the figure on the left presented below, all circles have center A and "same angle x" (shown in purple).

Please note that choosing convenient radii, we can create "circular sectors" with areas that are arbitrarily small, and arbitrarily large.
Consequence: the corresponding triangles (inside these circular sectors) can have areas that are also arbitrarily small, and arbitrarily large.

This is what we call a "GEOMETRIC BIFURCATION". This argument PROVES mathematically that (1) is insufficient!

(2) For any value of x (*) we have:

$$\left\{ \matrix{ \Delta \subset {\rm{circle}} \hfill \cr \Delta \ne {\rm{circle}} \hfill \cr} \right.\,\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,{S_\Delta } < {S_{{\rm{circle}}}} = \pi {\left( 1 \right)^2}\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\left\langle {{\rm{NO}}} \right\rangle$$

(*) We may implicitly assume that x is positive and less than 180 degrees. (Think about that!)

The correct answer is (B), indeed.

This solution follows the notations and rationale taught in the GMATH method.

Regards,
Fabio.
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Re: Is the area of the triangle drawn in circle A greater than π? &nbs [#permalink] 27 Sep 2018, 15:11
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