Question

ABC and BDE are two equilateral triangles such that D is the midpoint of BC. Ratio of these area of triangles ABC and BDE is 

(a) 2 : 1 

(b) 1 : 4 

(c) 1 : 2 

(d) 4 : 1

Answer 1

Give: ABC and BDE are two equilateral triangles 

Since, D is the midpoint of BC and BDE is also an equilateral triangle. 

Hence, E is also the midpoint of AB. 

Now, D and E are the midpoint of BC and AB. 

In a triangle, the line segment that joins midpoint of the two sides of a triangle is parallel to the third side and is half of it. 

DE ǁ CA and DE = 1/2 CA 

Now, in ∆ABC and ∆EBD 

∠BED = ∠BAC (corresponding angles) 

∠B = ∠B (common) 

By AA-similarity criterion 

∆ABC ~ ∆EBD 

If two triangles are similar, then the ratio of their areas is equal to the ratio of the squares of their corresponding sides.

Hence, the correct answer is option (d).