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The internal angles of a plane polygon are in AP. The smallest angle is 100o and the commondifference is 10o. Find the number of sides of the polygon.
- a)8
- b)9
- c)Either 8 or 9
- d)None of these
Correct answer is option 'C'. Can you explain this answer?
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Verified Answer
The internal angles of a plane polygon are in AP. The smallest angle i...
The sum of the interior angles of a polygon are multiples of 180 and are given by (n – 1) × 180
where n is the number of sides of the polygon. Thus, the sum of interior angles of a polygon would
be a member of the series: 180, 360, 540, 720, 900, 1080, 1260
The sum of the series with first term 100 and common difference 10 would keep increasing when
we take more and more terms of the series. In order to see the number of sides of the polygon, we
should get a situation where the sum of the series represented by 100 + 110 + 120… should
become a multiple of 180. The number of sides in the polygon would then be the number of terms
in the series 100, 110, 120 at that point.
If we explore the sums of the series represented by 100 + 110 + 120…
We realize that the sum of the series becomes a multiple of 180 for 8 terms as well as for 9 terms.
It can be seen in: 100 + 110 + 120 + 130 + 140 + 150 + 160 + 170 = 1080
Or 100 + 110 + 120 + 130 + 140 + 150 + 160 + 170 + 180 = 1260.
where n is the number of sides of the polygon. Thus, the sum of interior angles of a polygon would
be a member of the series: 180, 360, 540, 720, 900, 1080, 1260
The sum of the series with first term 100 and common difference 10 would keep increasing when
we take more and more terms of the series. In order to see the number of sides of the polygon, we
should get a situation where the sum of the series represented by 100 + 110 + 120… should
become a multiple of 180. The number of sides in the polygon would then be the number of terms
in the series 100, 110, 120 at that point.
If we explore the sums of the series represented by 100 + 110 + 120…
We realize that the sum of the series becomes a multiple of 180 for 8 terms as well as for 9 terms.
It can be seen in: 100 + 110 + 120 + 130 + 140 + 150 + 160 + 170 = 1080
Or 100 + 110 + 120 + 130 + 140 + 150 + 160 + 170 + 180 = 1260.
Most Upvoted Answer
The internal angles of a plane polygon are in AP. The smallest angle i...
Sum of interior angles of polygon=(n-2)x180 and sum of all interior angles of polygon of n sides in AP be found with a= 100 degrees and d= 10 and equated. after solving equation, we get =8,9 and hence c is answer
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The internal angles of a plane polygon are in AP. The smallest angle i...
Given:
- The internal angles of a plane polygon are in arithmetic progression (AP).
- The smallest angle is 100°.
- The common difference is 10°.
To find:
The number of sides of the polygon.
Solution:
Let's assume the number of sides of the polygon is "n".
Step 1: Finding the sum of angles
- The sum of the interior angles of an n-sided polygon is given by the formula: (n-2) * 180°.
- Since the angles are in AP, the sum of the angles can also be expressed as:
Sum = (n/2) * (2a + (n-1)d)
where a is the first term and d is the common difference.
Step 2: Finding the first term and common difference
- We are given that the smallest angle is 100°, which means it is the first term (a) of the AP.
- We are also given that the common difference (d) is 10°.
Step 3: Substituting the values
- Substituting the values in the sum formula:
(n/2) * (2*100 + (n-1)*10) = (n-2) * 180
Step 4: Simplifying the equation
- Expanding and simplifying the equation:
100n + 10n(n-1) = 360n - 720
10n^2 - 10n - 360n + 720 = 0
10n^2 - 370n + 720 = 0
Step 5: Solving the quadratic equation
- We can solve the quadratic equation using the quadratic formula:
n = (-b ± √(b^2 - 4ac)) / 2a
where a = 10, b = -370, and c = 720.
- Calculating the values:
n = (370 ± √((-370)^2 - 4*10*720)) / (2*10)
n = (370 ± √(136900 - 28800)) / 20
n = (370 ± √108100) / 20
n = (370 ± 329) / 20
- We have two possible values for n:
n1 = (370 + 329) / 20 = 699 / 20 = 34.95
n2 = (370 - 329) / 20 = 41 / 20 = 2.05
- Since the number of sides of a polygon cannot be a decimal or negative, we discard n2.
- Therefore, the number of sides of the polygon is n1 = 34.95.
Conclusion:
The number of sides of the polygon is approximately 34.95, which is not a whole number. Therefore, the correct answer is option 'C' - Either 8 or 9.
- The internal angles of a plane polygon are in arithmetic progression (AP).
- The smallest angle is 100°.
- The common difference is 10°.
To find:
The number of sides of the polygon.
Solution:
Let's assume the number of sides of the polygon is "n".
Step 1: Finding the sum of angles
- The sum of the interior angles of an n-sided polygon is given by the formula: (n-2) * 180°.
- Since the angles are in AP, the sum of the angles can also be expressed as:
Sum = (n/2) * (2a + (n-1)d)
where a is the first term and d is the common difference.
Step 2: Finding the first term and common difference
- We are given that the smallest angle is 100°, which means it is the first term (a) of the AP.
- We are also given that the common difference (d) is 10°.
Step 3: Substituting the values
- Substituting the values in the sum formula:
(n/2) * (2*100 + (n-1)*10) = (n-2) * 180
Step 4: Simplifying the equation
- Expanding and simplifying the equation:
100n + 10n(n-1) = 360n - 720
10n^2 - 10n - 360n + 720 = 0
10n^2 - 370n + 720 = 0
Step 5: Solving the quadratic equation
- We can solve the quadratic equation using the quadratic formula:
n = (-b ± √(b^2 - 4ac)) / 2a
where a = 10, b = -370, and c = 720.
- Calculating the values:
n = (370 ± √((-370)^2 - 4*10*720)) / (2*10)
n = (370 ± √(136900 - 28800)) / 20
n = (370 ± √108100) / 20
n = (370 ± 329) / 20
- We have two possible values for n:
n1 = (370 + 329) / 20 = 699 / 20 = 34.95
n2 = (370 - 329) / 20 = 41 / 20 = 2.05
- Since the number of sides of a polygon cannot be a decimal or negative, we discard n2.
- Therefore, the number of sides of the polygon is n1 = 34.95.
Conclusion:
The number of sides of the polygon is approximately 34.95, which is not a whole number. Therefore, the correct answer is option 'C' - Either 8 or 9.
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The internal angles of a plane polygon are in AP. The smallest angle is 100o and the commondifference is 10o. Find the number of sides of the polygon.a)8b)9c)Either 8 or 9d)None of theseCorrect answer is option 'C'. Can you explain this answer?
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