convex n-sided irregular polygon, largest int. angle is 160, smallest is 130.
- Thread starter lahsim
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What formulas have they given you for this? What rules or theorems? How far have you gotten in applying this information?In a convex n-sided irregular polygon , the largest interior angle is 160 while the smallest interior angle is 130. Find the greatest and least possible value of n. Answer is 14 and 6 but dont know how
Please be complete. Thank you!
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Start with the formula and the numbers they've given you. What is the largest value that the sum of the interior angles could be? What value of "n" does this suggest? What is the smallest value that the sum could be? What value of "n" does this suggest?
If you get stuck, please reply showing your efforts in answering the above questions. Thank you!
HallsofIvy
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I do not get that the smallest value of n is 6 and the largest value 14. I get that the smallest value of n is 8 and the largest is 18.
If the smallest angle is 130 degrees then the sum of all n angles must be larger than or equal to 130n.
If they largest angle is 160 degrees then the sum of all n angles must less than or equal to 160n.
So \(\displaystyle 130n\le (n- 2)180\le 160n\)
\(\displaystyle 130n\le 180n - 360\le 160n\)
\(\displaystyle 130n\le 180n- 360\) gives \(\displaystyle 50\ge 360\) so \(\displaystyle n\ge 36/5= 7.2\). Since n is an integer the smallest possible value of n is 8.
\(\displaystyle 180n- 360\le 160n\) so \(\displaystyle 20n\le 360\), \(\displaystyle n\le 18\).
If the smallest angle is 130 degrees then the sum of all n angles must be larger than or equal to 130n.
If they largest angle is 160 degrees then the sum of all n angles must less than or equal to 160n.
So \(\displaystyle 130n\le (n- 2)180\le 160n\)
\(\displaystyle 130n\le 180n - 360\le 160n\)
\(\displaystyle 130n\le 180n- 360\) gives \(\displaystyle 50\ge 360\) so \(\displaystyle n\ge 36/5= 7.2\). Since n is an integer the smallest possible value of n is 8.
\(\displaystyle 180n- 360\le 160n\) so \(\displaystyle 20n\le 360\), \(\displaystyle n\le 18\).