3. If ABC and EBC are two equilateral triangles such that Dis mid-point of BC, then the ratio of the areas of trianglesABC and BDE is:

Step 1: Understand the problem

We have two equilateral triangles ABC and EBC. D is the midpoint of BC. We need to find the ratio of the areas of triangles ABC and BDE.

Step 2: Recall the formula for the area of an equilateral triangle

The formula for the area of an equilateral triangle with side length a is (sqrt(3)/4)*a^2.

Step 3: Apply the formula to both triangles

Since D is the midpoint of BC, the side length of triangle BDE is half of the side length of triangle ABC. Therefore, the area of triangle BDE is (sqrt(3)/4)*(a/2)^2 = (sqrt(3)/16)*a^2.

Step 4: Find the ratio of the areas

The ratio of the areas of triangles ABC and BDE is ((sqrt(3)/4)*a^2) / ((sqrt(3)/16)*a^2) = 4.

So, the ratio of the areas of triangles ABC and BDE is 4:1.