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A regular polygon has an even number of sides. If the product of the length of its side and the distance between two opposite sides is 1/4 th of its area, find number of sides it has
(2016)
- a)6
- b)8
- c)20
- d)16
Correct answer is option 'D'. Can you explain this answer?
Most Upvoted Answer
A regular polygon has an even number of sides. If the product of the l...
Let the number of sides be 2n. Let the length of the side of S and the length of the perpendicular from the centre to each side be P. Since the number of sides is even, the opposite sides will be parallel and the distance between any two opposite sides is equal to 2P.
Also, area of the polygon (A) =
...(i)
According to question,
S(2P) = A/4 or SP = A/8
∴ (1) ⇒ A = n (A/8)
⇒ n = 8 or 2n = 16
∴ number of sides = 16.
Also, area of the polygon (A) =
...(i)According to question,
S(2P) = A/4 or SP = A/8
∴ (1) ⇒ A = n (A/8)
⇒ n = 8 or 2n = 16
∴ number of sides = 16.
Community Answer
A regular polygon has an even number of sides. If the product of the l...
Given, product of the length of its side and the distance between two opposite sides = 1/4th of its area. Let's assume the number of sides to be 'n'.
Area of the regular polygon = (n x s²)/(4tan(π/n)) where s is the length of a side.
Distance between two opposite sides = 2s sin(π/n)
Product of the length of its side and the distance between two opposite sides = 2s² sin(π/n)
1/4th of its area = (n x s²)/(16tan(π/n))
Equating both the expressions, we get:
2s² sin(π/n) = (n x s²)/(16tan(π/n))
Simplifying the above equation, we get:
8sin(π/n) = n tan(π/n)
We know that n is even, so let's substitute n = 2m in the above equation.
8sin(π/2m) = 2m tan(π/2m)
4sin(π/2m) = m tan(π/2m)
We can observe that the left-hand side of the above equation is less than or equal to 4, and the right-hand side is greater than or equal to m. Hence,
4sin(π/2m) <= m="">
We can also observe that the equality holds only when m=2, which means that n=2m=4.
Therefore, the regular polygon has 4 sides, and the answer is option 'D'.
Area of the regular polygon = (n x s²)/(4tan(π/n)) where s is the length of a side.
Distance between two opposite sides = 2s sin(π/n)
Product of the length of its side and the distance between two opposite sides = 2s² sin(π/n)
1/4th of its area = (n x s²)/(16tan(π/n))
Equating both the expressions, we get:
2s² sin(π/n) = (n x s²)/(16tan(π/n))
Simplifying the above equation, we get:
8sin(π/n) = n tan(π/n)
We know that n is even, so let's substitute n = 2m in the above equation.
8sin(π/2m) = 2m tan(π/2m)
4sin(π/2m) = m tan(π/2m)
We can observe that the left-hand side of the above equation is less than or equal to 4, and the right-hand side is greater than or equal to m. Hence,
4sin(π/2m) <= m="">
We can also observe that the equality holds only when m=2, which means that n=2m=4.
Therefore, the regular polygon has 4 sides, and the answer is option 'D'.
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A regular polygon has an even number of sides. If the product of the length of its side and the distance between two oppositesides is 1/4th of its area, find number of sides it has(2016)a)6b)8c)20d)16Correct answer is option 'D'. Can you explain this answer? for CAT 2025 is part of CAT preparation. The Question and answers have been prepared according to the CAT exam syllabus. Information about A regular polygon has an even number of sides. If the product of the length of its side and the distance between two oppositesides is 1/4th of its area, find number of sides it has(2016)a)6b)8c)20d)16Correct answer is option 'D'. Can you explain this answer? covers all topics & solutions for CAT 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for A regular polygon has an even number of sides. If the product of the length of its side and the distance between two oppositesides is 1/4th of its area, find number of sides it has(2016)a)6b)8c)20d)16Correct answer is option 'D'. Can you explain this answer?.
A regular polygon has an even number of sides. If the product of the length of its side and the distance between two oppositesides is 1/4th of its area, find number of sides it has(2016)a)6b)8c)20d)16Correct answer is option 'D'. Can you explain this answer? for CAT 2025 is part of CAT preparation. The Question and answers have been prepared according to the CAT exam syllabus. Information about A regular polygon has an even number of sides. If the product of the length of its side and the distance between two oppositesides is 1/4th of its area, find number of sides it has(2016)a)6b)8c)20d)16Correct answer is option 'D'. Can you explain this answer? covers all topics & solutions for CAT 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for A regular polygon has an even number of sides. If the product of the length of its side and the distance between two oppositesides is 1/4th of its area, find number of sides it has(2016)a)6b)8c)20d)16Correct answer is option 'D'. Can you explain this answer?.
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