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A regular polygon with 3x sides has interior angle 40degrees greater than that of one with x sides what is x
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rvkacademic

Answer:

[tex]\bullet\;\; x = 6[/tex]

Step-by-step explanation:

  • The sum of the angles of a polygon with [tex]n[/tex] sides is given by the formula:
    [tex]S = (n - 2) \times 180[/tex]
  • If it is a regular polygon of [tex]n[/tex] sides, each angle must be the same and the measure of each interior angle :
    [tex]\dfrac{(n-2) \cdot 180}{n}[/tex]
  • Given there are two regular polygons, one with x sides  and another with  [tex]3x[/tex] sides, let's write down the formula for the interior angles of each one.
  • For the polygon with [tex]x[/tex] sides:
    [tex]\displaystyle \textrm{Interior angle measure : } \dfrac{(x -2) \cdot 180}{x}[/tex]
  • For the polygon with [tex]3x[/tex] sides:
    [tex]\displaystyle \textrm{Interior angle measure : } \dfrac{(3x -2) \cdot 180}{3x}[/tex]
  • Given that the regular polygon with [tex]3x[/tex] sides has an interior angle 40 degrees greater than that of the polygon with [tex]x[/tex] sides, we can set up the following equation and solve for x

    [tex]\longrightarrow \dfrac{(3x -2) \cdot 180}{3x} =[/tex] [tex]\dfrac{(x -2) \cdot 180}{x} + 40\\\\[/tex]

    [tex]\\ \longrightarrow \dfrac{(3x -2) \cdot 180}{3x} - \dfrac{(x -2) \cdot 180}{x} = 40\\\\\\[/tex]   [tex]\longrightarrow \dfrac{(3x -2) \cdot 60}{x} - \dfrac{(x -2) \cdot 180}{x} = 40 \\\\\text{Multiply by x throughout}\\\\(3x -2) \cdot 60- (x -2) \cdot 180 = 40x\\\\\text{Simplify:}\\\\\rightarrow 180x - 120 - (180x - 360) = 40x\\\\\rightarrow180x-120-180x+360=40x\\\\\rightarrow-120+360=40x\rightarrow240=40x\rightarrow240/40=x\\\\\rightarrow6=x\\\\\rightarrow x=6[/tex]










mhanifa

x = 6

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To find the value of x in the context of a regular polygon, we begin with the fact that the sum of the interior angles of a polygon with n sides is given by the formula:

  • S = 180(n - 2) degrees

Each interior angle of a regular polygon (where all sides and angles are equal) can then be found by dividing this sum by the number of sides n.

So, the interior angle of a polygon with n sides is:

  • A = 180(n - 2)/n degrees

According to the given problem, the interior angle of a regular polygon with 3x sides is 40 degrees greater than that of one with x sides.

Therefore, we set up and solve the following equation:

  • 180(3x - 2)/(3x) = (180(x - 2)/x) + 40

By solving for x, we'll find the number of sides of the smaller polygon.

  • 180(3x - 2)/(3x) = (180(x - 2)/x) + 40
  • 180(3x - 2) = 3*180(x - 2) + 3x*40
  • 540x - 360 = 540x - 1080 + 120x
  • 120x = 720
  • x = 6

The smaller polygon has 6 sides and the larger one has 18 sides.

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