question_answer

Assertion (A): In a \[\Delta ABC,\] if \[\left. DE \right\|BC\] and intersects AB at D and AC at E, then \[\frac{AB}{AD}=\frac{AC}{AE}\].

Reason (R): If a line is drawn parallel to one side of a triangle to intersect the other two sides in distinct points, then these sides are divided in the same ratio.

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: A

Solution :

[a] Clearly, reason is true as it is Thales theorem.

Since, \[DE||BC,\] by Thales theorem, we have

\[\frac{AD}{DB}=\frac{AE}{BC}\Rightarrow \frac{DB}{AD}=\frac{EC}{AE}\]

\[\Rightarrow \,\,\,\,\,\,\,\,\,\,\,1+\frac{DB}{AD}=1+\frac{EC}{AE}\]

\[\Rightarrow \,\,\,\,\,\,\,\,\,\,\,\frac{AD+DB}{AD}=\frac{AE+EC}{AE}\]

\[\Rightarrow \,\,\,\,\,\,\,\,\,\,\,\frac{AB}{AD}=\frac{AC}{AE},\]which is true,

\[\therefore \] Assertion: True; Reason ; True and it is the correct explanation of assertion.