SOLUTION: One square is placed on top of another so that a regular octagon is formed, along with eight isosceles triangles each with its hypotenuse on a side of the octagon. If the perimeter

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Question 1148864: One square is placed on top of another so that a regular octagon is formed, along with eight isosceles triangles each with its hypotenuse on a side of the octagon. If the perimeter of the octagon is 20cm, find the perimeter of the star, in cm.
Diagram: https://imgur.com/a/C5NSnNK
Found 2 solutions by ankor@dixie-net.com, greenestamps:
Answer by ankor@dixie-net.com(22740) About Me  (Show Source):
You can put this solution on YOUR website!
One square is placed on top of another so that a regular octagon is formed, along with eight isosceles triangles each with its hypotenuse on a side of the octagon. If the perimeter of the octagon is 20cm, find the perimeter of the star, in cm.
:
Find the length of each side (c) of the octagon
c = 20%2F8
c = 2.5 cm, is the hypotenuse of the 8 triangles
Find the length of the sides (a) of the 8 triangles
2a^2 = 2.5^2
2a^2 = 6.25
a^2 = 6.25%2F2
a = sqrt%286.25%2F2%29
the 16 sides of the star
perimeter = 16%2Asqrt%286.25%2F2%29 = 28.284 cm

Answer by greenestamps(13202) About Me  (Show Source):
You can put this solution on YOUR website!


In each of the isosceles right triangles, the hypotenuse is 1/8 of the perimeter of the octagon, which is 20/8 = 2.5cm.

The sides of an isosceles right triangle are in the ratio 1:1:sqrt(2), so each leg of each isosceles triangle has a length 2.5/sqrt(2), or 1.25*sqrt(2).

The perimeter of the star is the combined perimeter of both legs of all the isosceles triangles, which is 16*(1.25*sqrt(2)) = 20*sqrt(2).

ANSWER: 20*sqrt(2) cm