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In the figure above, a circle is inscribed in a triangle.

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In the figure above, a circle is inscribed in a triangle. [#permalink] New post 16 Jan 2020, 10:21
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Question Stats:

88% (00:43) correct 12% (01:10) wrong based on 25 sessions
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#grrepepclub In the figure above, a circle is inscribed in a triangle..jpg
#grrepepclub In the figure above, a circle is inscribed in a triangle..jpg [ 15.24 KiB | Viewed 334 times ]



In the figure above, a circle is inscribed in a triangle.


Quantity A
Quantity B
The shaded area
The area of the circle



A)The quantity in Column A is greater.
B)The quantity in Column B is greater.
C)The two quantities are equal.
D)The relationship cannot be determined from the information given.

Kudos for the right answer and explanation
[Reveal] Spoiler: OA

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Re: In the figure above, a circle is inscribed in a triangle. [#permalink] New post 13 Feb 2020, 03:10
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As per the figure, no values have been prescribed. the shape of triangle may vary and hence the shaded area as well.
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Re: In the figure above, a circle is inscribed in a triangle. [#permalink] New post 13 Feb 2020, 23:39
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Carcass wrote:
Attachment:
#grrepepclub In the figure above, a circle is inscribed in a triangle..jpg



In the figure above, a circle is inscribed in a triangle.


Quantity A
Quantity B
The shaded area
The area of the circle



A)The quantity in Column A is greater.
B)The quantity in Column B is greater.
C)The two quantities are equal.
D)The relationship cannot be determined from the information given.

Kudos for the right answer and explanation



Logically it's clear that since no dimensions are given, the quantities cannot be compared.

In an acute triangle (equilateral), the area of the circle is greater than 50% of the triangle area.
As one angle keeps increasing at the cost of the other 2 angles - In an obtuse triangle (greater the value of the obtuse angle), the circle area keeps decreasing as a percent of the triangle area and is below 50%. ---- try sketching out different triangles to see for yourself


But let us try to analyse this.

Let's try to find the case where the circle area is exactly 50%:

Starting with an equilateral triangle of side s:
Inradius = 1/3 of the height = \(s/2\sqrt{3}\)
Area of circle = \(\pi*s²/12\)
Area of triangle = \(s²*\sqrt{3}/4\)
= \(s²*3\sqrt{3}/12\)
Here area of circle as a percent of the area of the triangle = 60.5%
Observe that this is way higher than 50%

Let's take a right triangle - 3-4-5 as the sides

Inradius = (3+4-5)/2 = 1
Basically: inradius = half of (sum of the legs - hypotenuse)
(Note: not explaining how this relation comes - try finding out yourself!)

Area of the triangle = 1/2 * 3 * 4 = 6
Area of the circle = \(\pi*1²\) = \(\pi\)

Here area of circle as a percent of the area of the triangle = 52.4%
Observe how it has come to almost 50%


Leaving it up to you guys to check out an obtuse triangle and see if it really falls below 50%

Note:
The above analysis is ONLY for academic interest. :-D

Answer D

Note: why am I checking 50%?
If the circle is more than 50% of the triangle, Qty B is greater; else it's Qty A
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Re: In the figure above, a circle is inscribed in a triangle.   [#permalink] 13 Feb 2020, 23:39
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In the figure above, a circle is inscribed in a triangle.

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