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12 Feb 2020, 20:43
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Difficulty:
85%
(hard)
Question Stats:
38% (01:49) correct
63%
(02:12)
wrong
based on 48
sessions
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Time
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Is the area of a certain circle greater than the area of a certain equilateral triangle?
1) The perimeter of the equilateral triangle is less than the circumference of the circle.
2) The circumference of the circle is less than the perimeter of the equilateral triangle.
1) The perimeter of the equilateral triangle is less than the circumference of the circle.
2) The circumference of the circle is less than the perimeter of the equilateral triangle.
12 Feb 2020, 23:46
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In this question, we need to establish the relationship between the side of the equilateral triangle and the radius of the circle to be able to answer the question.
The perimeter of an equilateral triangle is given by 3a where ‘a’ is the side of the equilateral triangle. The circumference of a circle is given by 2πr, where r is the radius of the circle.
The area of an equilateral triangle is given by\(\frac{ (√3 a^2)}{4}\).
The area of a circle is π\(r^2\).
Note that √3 = 1.732 and hence \(\frac{√3}{4}\) will be less than 1. As such, if you multiply this value with \(a^2\), you would be reducing the value of \(a^2\). Also note that π = 3.14 (approx.)
From statement I alone, 3a < 2πr. Simplifying, we have a < \(\frac{ 2πr }{ 3}\). From this, we can say that a is definitely less than r.
Therefore,
\(\frac{(√3 a^2)}{4}\) < π\(r^2\).
The area of the circle is definitely greater than the area of the equilateral triangle. Statement I alone is sufficient. Possible answer options are A or D.
From statement II alone, the circumference of the circle is less than the perimeter of the equilateral triangle.
That is, 2πr < 3a. This means, r<a. But this is insufficient to say whether the area of the circle will be greater than the area of the equilateral triangle or not.
For example, if we take a = 2π, we get r<3. For values of r<3 and a=2π, area of the equilateral triangle can be lesser or greater than the area of the circle depending on the value of r. Statement II alone is insufficient.
Answer option D can be eliminated. The correct answer option is A.
Hope that helps!
The perimeter of an equilateral triangle is given by 3a where ‘a’ is the side of the equilateral triangle. The circumference of a circle is given by 2πr, where r is the radius of the circle.
The area of an equilateral triangle is given by\(\frac{ (√3 a^2)}{4}\).
The area of a circle is π\(r^2\).
Note that √3 = 1.732 and hence \(\frac{√3}{4}\) will be less than 1. As such, if you multiply this value with \(a^2\), you would be reducing the value of \(a^2\). Also note that π = 3.14 (approx.)
From statement I alone, 3a < 2πr. Simplifying, we have a < \(\frac{ 2πr }{ 3}\). From this, we can say that a is definitely less than r.
Therefore,
\(\frac{(√3 a^2)}{4}\) < π\(r^2\).
The area of the circle is definitely greater than the area of the equilateral triangle. Statement I alone is sufficient. Possible answer options are A or D.
From statement II alone, the circumference of the circle is less than the perimeter of the equilateral triangle.
That is, 2πr < 3a. This means, r<a. But this is insufficient to say whether the area of the circle will be greater than the area of the equilateral triangle or not.
For example, if we take a = 2π, we get r<3. For values of r<3 and a=2π, area of the equilateral triangle can be lesser or greater than the area of the circle depending on the value of r. Statement II alone is insufficient.
Answer option D can be eliminated. The correct answer option is A.
Hope that helps!
19 May 2021, 04:33
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