question_answer

Assertion (A): In \[\Delta ABC,\] D and E are the points on sides AB and AC respectively such that \[\left. DE \right\|BC\] and \[\text{AD}:\text{DB}=\text{5}:\text{4}\].

Then \[ar(\Delta DFE):ar(\Delta CFB)=81:25\].

Reason (R): The ratio of the areas of two similar triangles is equal to the ratio of the squares of their corresponding sides.

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,  \[DE||BC\]

\[\Rightarrow \,\,\,\Delta ADE\tilde{\ }\Delta ABC\]

[By AAA similarity criterion]

\[\Rightarrow \,\,\,\,\,\,\,\,\,\,\frac{DE}{BC}=\frac{AD}{AB}\]                    .(1)

Given,   \[\frac{AD}{DB}=\frac{5}{4}\]

\[\Rightarrow \,\,\,\,\,\,\,\,\,\,\frac{AD}{AB}=\frac{5}{5+4}=\frac{5}{9}\]              .(2)

From eqs. (1) and (2), we have

\[\frac{DE}{BC}=\frac{5}{9}\]               .(3)

\[\Delta DEF\tilde{\ }\Delta CFB\]

\[[DE||BC\Rightarrow \angle 1=\angle 3\,\,\,and\,\,\angle 2=\angle 4]\]

\[\Rightarrow \,\,\,\,\frac{Area\,of\,\Delta DFE}{Area\,of\,\Delta CFB}=\frac{D{{E}^{2}}}{B{{C}^{2}}}={{\left( \frac{5}{9} \right)}^{2}}\]     [From (3)]

\[\Rightarrow \,\,\,\,\frac{Area\,of\,\Delta DFE}{Area\,of\,\Delta CFB}=\frac{25}{81}\]

\[\therefore \] Assertion: False; Reason: True.