question_answer

In a \[\Delta ABC,\] \[\left. DE \right\|BC.\]If \[DE=\frac{3}{2}BC\] and area of \[\Delta ABC=81c{{m}^{2}}\] then the area of \[\Delta ADE\]is:

A) \[\text{25 c}{{\text{m}}^{\text{2}}}\]

B) \[\text{36 c}{{\text{m}}^{\text{2}}}\]

C) \[\text{6 c}{{\text{m}}^{\text{2}}}\]

D) \[\text{5 c}{{\text{m}}^{\text{2}}}\]

Correct Answer: B

Solution :

[b] In \[\Delta ADE\] and \[\Delta ABC,\]

\[\angle DAE=\angle BAC\]                     (Common)

\[\angle ADE=\angle ABC\]

(Corresponding angles, as \[DE||BC\])

\[\therefore \,\,\,\,\Delta ADE\tilde{\ }\Delta ABC\](By AA similarity criterion)

Since, ratio of areas of two similar triangles is equal to the square of the ratio of any two corresponding sides,

\[\therefore \,\,\,\,\,\,\,\frac{ar(\Delta ADE)}{ar(\Delta ABC)}={{\left( \frac{DE}{BC} \right)}^{2}}\]

\[\Rightarrow \,\,\,\,\,\,\,\,\,\,\frac{ar(\Delta ADE)}{81}={{\left( \frac{\frac{2}{3}BC}{BC} \right)}^{2}}\]

\[\Rightarrow \,\,\,\,\,\,\,\,\,\,ar(\Delta ADE)=\frac{4}{9}\times 81=36c{{m}^{2}}\]