question_answer

The ratio of the area of the equilateral triangle described on the side of a square to the area of the equilateral triangle described on its diagonal is

A) 1 : 3     

B)  \[\sqrt{3}:4\]  

C)  1 : 2    

D)  1 : 4

Correct Answer: C

Solution :

Here, \[A{{C}^{2}}=2A{{B}^{2}}\]

As \[\Delta ABE\]and \[\Delta ABC\]are equiangular,

So,       \[\Delta ABE\sim \Delta ABC\]

[The ratio of the areas of two similar triangle is equal to the ratio of their corresponding sides]

\[\frac{\text{Area}\,\,\text{of}\,\,\Delta ABE}{\text{Area}\,\,\text{of}\,\,\Delta ACF}=\frac{A{{B}^{2}}}{A{{C}^{2}}}=\frac{A{{B}^{2}}}{2A{{B}^{2}}}=\frac{1}{2}\]