Question

ABC and BDE are two equilateral triangles such that D is the midpoint of BC. The ratio of the areas of the triangles ABC and BDE is:

• A. \( 2 : 1 \)
• B. \( 1 : 2 \)
• C. \( 4 : 1 \)
• D. \( 1 : 4 \)

Hint:

The area of an equilateral triangle is proportional to the square of its side length. So, if the side of one triangle is half of another, the area will be one-fourth.

Answer 1

The Correct Option is D

We are given two equilateral triangles \( ABC \) and \( BDE \), and \( D \) is the midpoint of \( BC \). The area of an equilateral triangle is proportional to the square of the length of its side.
Step 1: Understanding the relationship between the triangles.
Let the side length of triangle \( ABC \) be \( s \). Since \( D \) is the midpoint of \( BC \), the side length of triangle \( BDE \) is half of that of triangle \( ABC \), i.e., \( \frac{s}{2} \).
Step 2: Using the formula for the area of an equilateral triangle.
The area of an equilateral triangle is proportional to the square of the side length. Therefore, the ratio of the areas of triangles \( ABC \) and \( BDE \) is the ratio of the squares of their side lengths: \[ \text{Area ratio} = \left( \frac{\text{side of } ABC}{\text{side of } BDE} \right)^2 = \left( \frac{s}{\frac{s}{2}} \right)^2 = (2)^2 = 4 \]
Step 3: Conclusion.
Therefore, the ratio of the areas of triangle \( ABC \) to triangle \( BDE \) is \( 4 : 1 \).