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in Triangles by (35.6k points)
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In ΔABC, D and E are midpoints of AB and AC then DE : BC 

(A) 4 : 1 

(B) 1 : 1 

(C) 1 : 2 

(D) 2 : 1

2 Answers

+1 vote
by (57.0k points)
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Best answer

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

+1 vote
by (34.5k points)

Correct option is: (C) 1 : 2

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