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Two regular polygons are such that the ratio between their number of sides is 1:2ad the ratio of measures of their interior angles is 3:4.find the number of sides of each polygon?
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Two regular polygons are such that the ratio between their number of s...
Problem: Find the number of sides of two regular polygons given the ratio of their number of sides is 1:2 and the ratio of measures of their interior angles is 3:4.
Solution:
Let the number of sides of the two polygons be n and 2n, respectively.
Ratio of number of sides:
Since the ratio of their number of sides is 1:2, we have:
n:2n = 1:2
Cross-multiplying, we get:
n x 2 = 2n x 1
Simplifying, we get:
n = 2
Therefore, the polygons have 2 sides and 4 sides, respectively.
Ratio of measures of interior angles:
The interior angle of a regular n-gon is given by:
180(n-2)/n
Therefore, the interior angle of the first polygon, with n sides, is:
180(n-2)/n
And the interior angle of the second polygon, with 2n sides, is:
180(2n-2)/(2n)
The ratio of their measures is given to be 3:4. Therefore, we have:
[180(n-2)/n] : [180(2n-2)/(2n)] = 3:4
Cross-multiplying, we get:
[180(n-2)/n] x 4 = [180(2n-2)/(2n)] x 3
Simplifying, we get:
12n - 24 = 6n
Solving for n, we get:
n = 4
Therefore, the first polygon has 4 sides and the second polygon has 8 sides.
Final Answer:
The two regular polygons have 4 sides and 8 sides, respectively.
Solution:
Let the number of sides of the two polygons be n and 2n, respectively.
Ratio of number of sides:
Since the ratio of their number of sides is 1:2, we have:
n:2n = 1:2
Cross-multiplying, we get:
n x 2 = 2n x 1
Simplifying, we get:
n = 2
Therefore, the polygons have 2 sides and 4 sides, respectively.
Ratio of measures of interior angles:
The interior angle of a regular n-gon is given by:
180(n-2)/n
Therefore, the interior angle of the first polygon, with n sides, is:
180(n-2)/n
And the interior angle of the second polygon, with 2n sides, is:
180(2n-2)/(2n)
The ratio of their measures is given to be 3:4. Therefore, we have:
[180(n-2)/n] : [180(2n-2)/(2n)] = 3:4
Cross-multiplying, we get:
[180(n-2)/n] x 4 = [180(2n-2)/(2n)] x 3
Simplifying, we get:
12n - 24 = 6n
Solving for n, we get:
n = 4
Therefore, the first polygon has 4 sides and the second polygon has 8 sides.
Final Answer:
The two regular polygons have 4 sides and 8 sides, respectively.
Community Answer
Two regular polygons are such that the ratio between their number of s...
Let the number of sides of the regular polygon with the fewer number of sides be x, and y be the number of sides of the polygon with the greater number of sides.
xy=12 solving for y: y=2x
The measure of an interior angle of a regular polygon (I) is:
I=(n−2)180n
(x−2)180x(y−2)180y=34
Simplifying:
y(x−2)x(y−2)=34
Substituting for y:
2x(x−2)x(2x−2)=34
Cross multiplying and distributing:
8x2−16x=6x2−6x
Combine like terms:
2x2−10x=0
Factor:
2x(x−5)=0
Therefore, x = 0 or x = 5.
Eliminate x = 0, thus, x = 5 and y = 10, or the number of sides of the two regular polygons are 5 and 10.
xy=12 solving for y: y=2x
The measure of an interior angle of a regular polygon (I) is:
I=(n−2)180n
(x−2)180x(y−2)180y=34
Simplifying:
y(x−2)x(y−2)=34
Substituting for y:
2x(x−2)x(2x−2)=34
Cross multiplying and distributing:
8x2−16x=6x2−6x
Combine like terms:
2x2−10x=0
Factor:
2x(x−5)=0
Therefore, x = 0 or x = 5.
Eliminate x = 0, thus, x = 5 and y = 10, or the number of sides of the two regular polygons are 5 and 10.
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Two regular polygons are such that the ratio between their number of sides is 1:2ad the ratio of measures of their interior angles is 3:4.find the number of sides of each polygon?
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Two regular polygons are such that the ratio between their number of sides is 1:2ad the ratio of measures of their interior angles is 3:4.find the number of sides of each polygon? for Class 7 2024 is part of Class 7 preparation. The Question and answers have been prepared according to the Class 7 exam syllabus. Information about Two regular polygons are such that the ratio between their number of sides is 1:2ad the ratio of measures of their interior angles is 3:4.find the number of sides of each polygon? covers all topics & solutions for Class 7 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Two regular polygons are such that the ratio between their number of sides is 1:2ad the ratio of measures of their interior angles is 3:4.find the number of sides of each polygon?.
Two regular polygons are such that the ratio between their number of sides is 1:2ad the ratio of measures of their interior angles is 3:4.find the number of sides of each polygon? for Class 7 2024 is part of Class 7 preparation. The Question and answers have been prepared according to the Class 7 exam syllabus. Information about Two regular polygons are such that the ratio between their number of sides is 1:2ad the ratio of measures of their interior angles is 3:4.find the number of sides of each polygon? covers all topics & solutions for Class 7 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Two regular polygons are such that the ratio between their number of sides is 1:2ad the ratio of measures of their interior angles is 3:4.find the number of sides of each polygon?.
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