Answer
See the answer below.
Work Step by Step
$B.O.=N.\ Shared\ pairs\div N.\ links$
$NO_2^+$: $4/2=2$
$NO_2^-$: $3/2=1.5$
Since bond length decreases with bond order, the 110 pm is from $NO_2^+$ and the 124 pm is from $NO_2^-$.
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