question_answer

ABC and BDE are two equilateral triangles such that D is the mid-point of SC. Ratio of the areas of triangles ABC and BDE is:

A) 2:1

B) 1:2

C) 4:1

D) 1:4

Correct Answer: C

Solution :

[c]: Given, ABC and BDE are two equilateral triangles.

\[\Rightarrow \,\,\Delta ABC\]and \[\Delta BDE\] are equiangular.

\[\therefore \,\,\,\Delta ABC\tilde{\ }\Delta BDE\]

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

\[\therefore \,\,\frac{ar(ABC)}{ar(\Delta BDE)}={{\left( \frac{AB}{BD} \right)}^{2}}={{\left( \frac{BC}{BD} \right)}^{2}}={{\left( \frac{2BD}{BD} \right)}^{2}}\]

( Given, D is mid-point of BC \[\therefore \,BC=2BD\])

\[\Rightarrow \,\,\,\,\,\frac{ar\,(\Delta ABC)}{ar(\Delta BDE)}=\frac{4}{1}\]