The diagram shows two congruent regular pentagons and part of a regular n-sided polygon A. Two sides of each of the regular pentagons and two side of A meet at the pint P. Calculate the value of n.​

Answer:

n=10

Step-by-step explanation:

For a regular polygon with n sides, each interior angle, x, is given by

[tex]x=180\frac{n-2}{n}[/tex]

This can be solved for n

[tex]n=\frac{360}{180-x}[/tex]

For the pentagons

x(5)=180(3/5)=108 degrees.

The angles around P sum to 360, so

2(x(5))+x(A)=360

x(A)=360-2(108)=144.

Substitue in the equation for n.

n=360/(180-144)=10

The number of sides in this polygon is 10.

For any polygon, the sum of interior angles is given as,

[tex]\bold{Sum\ of\ interior\ angles= {(n-2)}\times {180^o}}[/tex]

where,

n = number of sides,

We know that for a pentagon, the number of sides is 5.

thus, substituting the values in the sum of interior angle formula we get,

[tex]{Sum\ of\ interior\ angles= {(n-2)}\times {180^o}}[/tex]

[tex]{Sum\ of\ interior\ angles= {5-2}\times{180^o}}\\= 3 \times {180^o} = 540^o[/tex]

therfore, each angle in a regular pentagon,

[tex]\dfrac{Sum\ of\ interior\ angles}{Number\ of\ Sides} = \dfrac{540^o}{5} = 108^o[/tex],

Let's assume that polygon A is having x number of sides.

At point P, the sum of the angles must be 360°.

The sum of the angles at point P

          = interior angle of the first pentagon

          + interior angle of the second pentagon

          + interior angle of the polygon A with x sides

Substituting the values,

360° = 108° + 108° + z

z = 360° - 108° - 108°

z = 144°

Thus, the interior angle of the x sided polygon is z = 144°.

For a regular polygon, all the interior angles are equal,

so, for a polygon with sides x.

the sum of interior angle = measure of interior angle x Number of sides

                                         = [tex]144^o \times x[/tex]

Also, we know

[tex]{Sum\ of\ interior\ angles= {(n-2)}\times {180^o}}[/tex]

Substituting, the values for an x number of side  polygon,

[tex]{Sum\ of\ interior\ angles= {(n-2)}\times {180^o}}\\{144^o \times x = {(x-2)}\times {180^o}}\\144x = 180x - 360\\360 = 180x- 144x\\x = \dfrac{360}{36}\\x = 10[/tex]

Hence, the number of sides in this polygon is 10.

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