Question 1194423: A regular pologon has it exterior angle being 100 degrees less than the interior angle,find the number of sides of the pologon.
Found 3 solutions by math_tutor2020, Alan3354, MathTherapy:
Answer by math_tutor2020(3817)   (Show Source):
You can put this solution on YOUR website!
Definition: A regular polygon has all sides equal to one another. Also, the angles are equal to one another.
Examples:- An equilateral triangle is a regular polygon with 3 equal sides of any length of your choosing. Each angle is 60 degrees.
- A square is a regular polygon with 4 equal sides of your choosing. Each angle is 90 degrees.
n = number of sides of the regular polygon
i = interior angle
E = exterior angle
We're told that "exterior angle being 100 degrees less than the interior angle", so,
Exterior angle = (interior angle) - 100
E = i - 100
If a regular polygon has n sides, then the interior angle (i) can be determined through this formula
i = 180(n-2)/n
The exterior angle (E) can be determined like so
E = 360/n
Apply substitution and solve for n.
E = i - 100
360/n = 180(n-2)/n - 100
360 = 180(n-2) - 100n
360 = 180n-360 - 100n
360 = 80n-360
360+360 = 80n
720 = 80n
n = 720/80
n = 9
This regular polygon has 9 sides.
This is a regular nonagon.
Note: in step 3, I multiplied both sides by n to clear out the denominators.
If this regular polygon has 9 sides, then,
E = 360/n = 360/9 = 40 degrees
i = 180(n-2)/n = 180(9-2)/9 = 140 degrees
We can see those values satisfy E = i - 100 to help confirm the answer.
Answer: 9 sides
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Edit:
I'm realizing there's an alternative method
E = i - 100 is one equation we can form
i+E = 180 is another equation
Apply substitution and solve
i+E = 180
i+i-100 = 180
2i-100 = 180
2i = 180+100
2i = 280
i = 280/2
i = 140
and
E = i - 100
E = 140-100
E = 40
Then we can say
n = 360/E = 360/40 = 9
There are 9 sides to the polygon.
Answer by Alan3354(69443)  (Show Source):
You can put this solution on YOUR website! A regular pologon [sic] has it exterior angle being 100 degrees less than the interior angle, find the number of sides of the pologon [sic].
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polygon, not pologon
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Ext + Int = 180 degs
(180 - 100)/2 = 40 degs (exterior angles)
360/40 = 9 sides
Answer by MathTherapy(10556)  (Show Source):
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