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The area of the circle above, with center C, is

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The area of the circle above, with center C, is [#permalink] New post   12 Oct 2020, 07:33
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The area of the circle above, with center C, is \(36\pi\) , and \(x > 90\)


Quantity A
Quantity B
The area of the triangle
18


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.

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Re: The area of the circle above, with center C, is [#permalink] New post   Updated on: 15 Oct 2020, 03:42
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The area of any triangle with a x° angle in the domain of 90° < x < 180°, is less than the area of a triangle with x=90°. Having said that, we can calculate the area of a 90 degress triangle:

Area of the circle
\(Á_{circle} = \pi*r = 36\pi\)
\(r = 6\)
but, we know that the area of a triangle is given by:
\(Á_{triangle} = \frac{b*h}{2}\)
and, because we have a isoseles, the area is:
\(Á_{triangle} = \frac{r*r}{2} = \frac{6*6}{2}\)
\(Á_{triangle} = 18\)
but, x is MORE THAN 90°, therefore, we are completely sure that the area of any triangle in that domain must be less than 18.

Option B

Originally posted by cote on 13 Oct 2020, 19:23.
Last edited by cote on 15 Oct 2020, 03:42, edited 1 time in total.
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Re: The area of the circle above, with center C, is [#permalink] New post   15 Oct 2020, 02:28
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cote wrote:
The area of any triangle with a x° angle in the domain of 90° < x < 180°, is less than the area of a triangle with x=90°. Having said that, we can calculate the area of a 90 degress triangle:

Area of the circle
\(Á_{circle} = \pi*r = 36\pi\)
\(r = 6\)
but, we know that the area of a triangle is given by:
\(Á_{triangle} = \frac{b*h}{2}\)
and, because we have a isoseles, the area is:
\(Á_{triangle} = \frac{r*r}{2} = \frac{6*6}{2}\)
\(Á_{triangle} = 18\)
but, x is LESS THAN 90°, therefore, we are completely sure that the area of any triangle in that domain must be less than 18.

Option B


it is mentioned in question that
x is > 90.,
therefore ans should be A
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Re: The area of the circle above, with center C, is [#permalink] New post   15 Oct 2020, 03:41
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fabiha22 wrote:
cote wrote:
The area of any triangle with a x° angle in the domain of 90° < x < 180°, is less than the area of a triangle with x=90°. Having said that, we can calculate the area of a 90 degress triangle:

Area of the circle
\(Á_{circle} = \pi*r = 36\pi\)
\(r = 6\)
but, we know that the area of a triangle is given by:
\(Á_{triangle} = \frac{b*h}{2}\)
and, because we have a isoseles, the area is:
\(Á_{triangle} = \frac{r*r}{2} = \frac{6*6}{2}\)
\(Á_{triangle} = 18\)
but, x is LESS THAN 90°, therefore, we are completely sure that the area of any triangle in that domain must be less than 18.

Option B


it is mentioned in question that
x is > 90.,
therefore ans should be A


Sorry, there is a typo in the last sentence (the option is still B). In the beginning, I exposed the opposite:

The area of any triangle with a x° angle in the domain of 90° < x < 180°, is less than the area of a triangle with x=90°
,

therefore, it should be: "x is MORE THAN 90°"

thanks!
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Re: The area of the circle above, with center C, is [#permalink] New post   15 Oct 2020, 23:29
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Quote:
The area of the circle above, with center C, is 36π, and x>90

Step 1: Understanding the question
Area of the circle = π r^2 = 36π
r = 6

Area of a triangle is maximum when the triangle is a right angle triangle.

Step 2: Calculation
When x = 90, area of the triangle = 1/2 * 6 * 6 = 18
As x > 90, area of the triangle is less than 18

B is correct
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The area of the circle above, with center C, is [#permalink] New post   Updated on: 12 Jan 2021, 10:41
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All of the answers is write but with no proof. How did you know that the area is less when the angle is more than 90 degree?
Actually, the formula for the area of an isosceles triangle with sides a is a^2*Sinx/2. this is the reason.

Originally posted by amirbehani on 10 Jan 2021, 11:22.
Last edited by amirbehani on 12 Jan 2021, 10:41, edited 1 time in total.
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Re: The area of the circle above, with center C, is [#permalink] New post   10 Jan 2021, 22:56
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First step is to find the radius of circle.
pi*r^2 = 36*pi
r=6

Now we know area of triangle is 0.5*Base*Height.
If x had been 90, the area would have been 0.5*6*6=18, simply because the height and base are given for this particular scenario.
Now if we increase the angle x, the height reduces, thus if base value remains same, and height is decreasing, the area will also decrease.

Thus Qb>Qa
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Re: The area of the circle above, with center C, is [#permalink] New post   12 Jan 2021, 10:50
Quiqq wrote:
First step is to find the radius of circle.
pi*r^2 = 36*pi
r=6

Now we know area of triangle is 0.5*Base*Height.
If x had been 90, the area would have been 0.5*6*6=18, simply because the height and base are given for this particular scenario.
Now if we increase the angle x, the height reduces, thus if base value remains same, and height is decreasing, the area will also decrease.

Thus Qb>Qa


You are completely wrong. Math is the language of mathematic formulas not explanation. Moreover your explanation is also wrong, because when the angle x becomes greater, the base value also increases. you cannot say " if the base value remains the same".
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Re: The area of the circle above, with center C, is [#permalink] New post   15 Dec 2021, 07:13
amirbehani wrote:
Quiqq wrote:
First step is to find the radius of circle.
pi*r^2 = 36*pi
r=6

Now we know area of triangle is 0.5*Base*Height.
If x had been 90, the area would have been 0.5*6*6=18, simply because the height and base are given for this particular scenario.
Now if we increase the angle x, the height reduces, thus if base value remains same, and height is decreasing, the area will also decrease.

Thus Qb>Qa


You are completely wrong. Math is the language of mathematic formulas not explanation. Moreover your explanation is also wrong, because when the angle x becomes greater, the base value also increases. you cannot say " if the base value remains the same".


Sir, I believe you are wrong. Upon increasing the angle at the center above 90 degrees, the height does reduce, as the base will always remain same as the radius.
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Re: The area of the circle above, with center C, is [#permalink] New post   14 Aug 2024, 09:54
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Re: The area of the circle above, with center C, is [#permalink]
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