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26 Feb 2022, 23:35
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The measures of the interior angles of a convex polygon of n sides are in arithmetic progression. If the common difference is 5* and the largest angle is 160*, then n equals what?
A. 9
B. 10
C. 12
D. 16
E. 32
A. 9
B. 10
C. 12
D. 16
E. 32
27 Feb 2022, 14:53
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The measures of the interior angles of a convex polygon of n sides are in arithmetic progression. If the common difference is 5* and the largest angle is 160*, then n equals what?
A. 9
B. 10
C. 12
D. 16
E. 32
A. 9
B. 10
C. 12
D. 16
E. 32
Breaking Down the Info:
The number of degrees in total for a polygon of n sides is \(180*(n - 2)\).
We are given the current polygon has angles of a sequence of n sides and the angles have a common difference of 5. If the biggest angle is 160 then the smallest angle is \(160 - 5(n - 1) = 165 - 5n\). Therefore the median/mean is \(\frac{165 - 5n + 160}{2} = \frac{325 - 5n}{2}\), which makes the sum of angles \(n*\frac{325 - 5n}{2} = 180*(n - 2)\)
Dividing both sides by 5 gives us \(65n - n^2 = 72(n - 2)\) and \(n^2 + 7n - 144 = 0\). We can break 144 = 16*9, so \((n + 16)(n - 9) = 0\) and n must be 9.
Answer: A
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