Question

Corners are cut from an equilateral triangle to produce a regular convex hexagon as shown in the figure above. The ratio of the area of the regular convex hexagon to the area of the original equilateral triangle is 

• A. 2 : 3
• B. 3 : 4
• C. 4 : 5
• D. 5 : 6

Hint:

To solve problems involving geometric shapes like triangles and hexagons, focus on the proportionality of areas. Cutting the corners of a triangle to form a hexagon reduces the area in a fixed proportion.

Answer 1

The Correct Option is A

The problem involves cutting the corners of an equilateral triangle to form a regular convex hexagon. We need to find the ratio of the area of the regular convex hexagon to the area of the original equilateral triangle. Step 1: Understand the geometry of the problem.
When corners are cut off an equilateral triangle, the resulting shape is a regular convex hexagon. The key to solving this problem is recognizing that the area of the regular hexagon is proportional to the area of the equilateral triangle from which it is formed. Step 2: Calculate the area of the equilateral triangle.
The area of an equilateral triangle with side length \( a \) is given by the formula: \[ A_{\text{triangle}} = \frac{\sqrt{3}}{4} a^2 \] Step 3: Calculate the area of the regular hexagon.
The regular hexagon formed by cutting the corners of the equilateral triangle will have a side length that is a fraction of the side length of the equilateral triangle. After cutting off the corners, the remaining area is that of the regular hexagon. The area of the hexagon can be calculated using the formula for the area of a regular hexagon with side length \( s \): \[ A_{\text{hexagon}} = \frac{3\sqrt{3}}{2} s^2 \] However, for this case, the area of the hexagon is proportional to the area of the original triangle, and the proportionality constant comes out to be \( \frac{2}{3} \). Step 4: Find the ratio of areas.
The ratio of the area of the regular hexagon to the area of the original equilateral triangle is: \[ \frac{A_{\text{hexagon}}}{A_{\text{triangle}}} = \frac{2}{3} \] Thus, the ratio is \( 2 : 3 \), which corresponds to option (A). Final Answer: 2 : 3