Correct option is (C) 1 : 2
Given that D is mid-point of AB & E is mid-point of AC.
\(\therefore AD=\frac12AB\;and\;AE=\frac12AC\)
\(\Rightarrow\frac{AD}{AB}=\frac{1}{2}\;and\;\frac{AE}{AC}=\frac{1}{2}\)
\(\Rightarrow\frac{AD}{AB}=\frac{AE}{AC}\) ________________(1)
Now in triangle \(\triangle ADE\;and\;\triangle ABC,\)
\(\angle DAE=\angle BAC\) (Common angle)
and \(\frac{AD}{AB}=\frac{AE}{AC}\) (From (1))
\(\therefore\) \(\triangle ADE\sim\triangle ABC\) (By SAS similarity rule)
\(\therefore\frac{DE}{BC}=\frac{AD}{AB}=\frac{AE}{AC}=\frac12\)
\(\therefore\) DE : BC = 1 : 2