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This problem illustrates the fact that two functions can be very close: $|f(x)-g(x)|<\epsilon$ for all $x\in [0,1]$, but their derivatives can still be far apart, $|f'(x)-g'(x)|>c$ for some constant $c>0$. In our case, let $x=a(t),y=b(t),0\le t\le 1$ and $x=c(t),y=d(t), 0\le t\le 1$ be the parametrizations of the two curves. By smoothing the corners, we may assume that both are smooth. $$ \|(a(t),b(t)\|\approx \|(c(t),d(t)\|$$ does not imply $$ \|(a'(t),b'(t)\|\approx \|(c'(t),d'(t)\|$$ Therefore $\int_0^1 \|(a'(t),b'(t)\| dt$ need not be close to $\int_0^1 \|(c'(t),d'(t)\| dt.$ Here $\|(x,y\|$ denotes $\sqrt{x^2+y^2}$.