Answer:
Part 1) The length of the apothem is 13.32'
Part 2) The perimeter of the decagon is 86.5'
Step-by-step explanation:
we know that
A regular decagon has 10 equal sides and 10 equal interior angles
A regular decagon can be divided into 10 congruent isosceles triangle
(they are isosceles since their two sides are the radii of the polygon and the unknown side is the side of the polygon)
The vertex angle of each isosceles triangle is equal to
To find out the side length of the decagon, we can use the law of cosines
so
where
c is the length side of decagon
a and b are the radii
we have
substitute the values
To fin out the perimeter of decagon multiply the length side by 10
so
To find out the apothem we can apply the Pythagorean Theorem in one isosceles triangle
see the attached figure to better understand the problem
substitute the given values
solve for a
