Equilateral Triangle and Inscribed Square

To solve this problem, let's break it down into the following steps:

1. Understanding the Problem: We are given an equilateral triangle and a square. The triangle is circumscribed, meaning that the vertices of the triangle lie on a circle with radius r. The square is inscribed, meaning that its four vertices touch the circle. We need to find the ratio of the areas of the triangle and the square.

2. Finding the Area of the Triangle: The area of an equilateral triangle can be calculated using the formula A = (sqrt(3) / 4) * a^2, where A is the area and a is the length of the side of the triangle. Since the triangle is circumscribed, the length of each side is equal to the diameter of the circle, which is 2r. Therefore, the area of the equilateral triangle is A_triangle = (sqrt(3) / 4) * (2r)^2 = (sqrt(3) / 4) * 4r^2 = sqrt(3) * r^2.

3. Finding the Area of the Square: The square is inscribed in the circle, which means that the diagonal of the square is equal to the diameter of the circle, which is 2r. Let's denote the side length of the square as s. Using the Pythagorean theorem, we can find the relationship between the diagonal and the side length of the square: s^2 + s^2 = (2r)^2. Simplifying this equation, we get 2s^2 = 4r^2, which further simplifies to s^2 = 2r^2. Therefore, the area of the square is A_square = s^2 = 2r^2.

4. Calculating the Ratio: Now that we have the areas of both the triangle and the square, let's calculate the ratio (t/s): (t/s) = A_triangle / A_square = (sqrt(3) * r^2) / (2r^2) = sqrt(3) / 2.

Conclusion: The ratio of the areas of the equilateral triangle (t) and the inscribed square (s) is sqrt(3) / 2.

Bhai apun ko bhi yahi doubt ha ..hahahabaja lolol