Question

The difference between the interior and the exterior angle of a regular polygon is 100degree . find the number of sides of the polygon. ?

Answer 1

The polygon has 9 sides

Explanation:

What information do we know and how do we use it to model this situation?

`color(green)("Let the number of sides be "n)`
`color(green)("Let internal angle be "color(white)(.......)A_i`
`color(green)("Let external angle be "color(white)(.......)A_e`
Assumption: External angle less than internal angle `color(green)(-> A_e < A_i)`

Thus `color(green)(A_i - A_e>0 => A_i - A_e=100`

Not that `sum " is: the sum of"`

`color(brown)("Known: "underline("Sum of internal angles is")color(white)(..)color(green)((n-2)180))`

So `color(green)(sumA_i = (n-2)180................................(1))`

`color(brown)("Known:"underline(" Sum of external angles is")color(white)(..)color(green)(360^0))`

So `color(green)(sumA_e=360 ..............................................(2))`

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`color(blue)("Equation (1) - Equation (2)")`

`sum (A_i-Ae)= (n-2)180 -360`

But also `sum (A_i-Ae)= sum "difference"`

There are `n` sides each with a difference of `100^0`
So `sum "difference" = 100n` giving:

`color(green)(sum (A_i-Ae) = 100n = (n-2)180 -360.................(3))`

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`color(blue)("Collecting like terms")`

`100n = 180n - 360 - 360`

`80n =720`

`n=720/80 = 9`