question_answer

Assertion (A): In a \[\Delta ABC,\] D and E are points on sides AB and AC respectively such that \[\text{BD}=\text{CE}\]. If \[\angle B=\angle C,\] then DE is not parallel to BC.

Reason (R): If a line divides any two sides of a triangle in the same ratio, then the line must be parallel to the third side.

A) Both assertion (A) and reason (R) are true and reason (R) is the correct explanation of assertion (A).

B) Both assertion (A) and reason (R) are true but reason (R) is not the correct explanation of assertion (A).

C) Assertion (A) is true but reason (R) is false.

D) Assertion (A) is false but reason (R) is true.

Correct Answer: D

Solution :

[d] Clearly, reason is true.

Now, In \[\Delta ABC,\] we have \[\angle B=\angle C\]

\[\Rightarrow \,\,\,\,\,\,\,\,\,\,\,\,AC=AB\]

[Sides opposite to equal angles are equal]

\[\Rightarrow \,\,\,\,\,\,\,\,\,\,\,AE+EC=AD+DB\]

\[\Rightarrow \,\,\,\,\,\,\,\,\,\,\,AE+CE=AD+BD\]

\[\Rightarrow \,\,\,\,\,\,\,\,\,\,\,AE+CE=AD+CE\][\[BD=CE\]Given)]

\[\Rightarrow \,\,\,\,\,\,\,\,\,\,\,AE=AD\]

Thus, we have

\[AD=AE\]

and      \[BD=CE\]

\[\therefore \,\,\,\,\,\frac{AD}{BD}=\frac{AE}{CE}\,\,\,\Rightarrow \,\,\,\frac{AD}{DB}=\frac{AE}{EC}\]

\[\Rightarrow \,\,\,\,\,\,\,\,\,\,DE||BC\]

AD^AE    AD.AE.

\[\therefore \]  Assertion; False; Reason ; True,