Answer: 238
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How to get this answer
Focus on the top most sides of 4 and 28, which are on the smaller and larger octagons respectively.
The jump from 4 to 28 has us multiply by 7
4*7 = 28
The other sides will follow the same pattern of "multiply by 7"
3*7 = 21
7*7 = 49
6*7 = 42
See the diagram below
After you know all the 7 other sides of the larger octagon, you add them all up
28+28+21+21+49+28+21+42 = 238
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With a lot of math problems, there's often an alternate route which is much faster. In this case, we can multiply the perimeter of the smaller octagon by the scale factor 7 to get the same answer
(smaller perimeter)*(scale factor) = (larger perimeter)
34*7 = 238
so the larger perimeter is 238
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Why does this work? Let's say the sides of the smaller octagon are: a,b,c,d,e,f,g,h
adding up those sides gets us a+b+c+d+e+f+g+h
now imagine multiplying all those sides by the same number 7
a becomes 7a, b becomes 7b, and so on
The new larger perimeter is 7a+7b+7c+7d+7e+7f+7g+7h which we can factor out that 7 and we get 7*(a+b+c+d+e+f+g+h). The stuff inside the parenthesis is exactly the old smaller perimeter we found earlier, so therefore we have
new perimeter = 7*(old perimeter)