The large red equilateral triangle has sides of 8x units. The midpoints of the red triangle are joined to form the blue triangle. The midpoints of the blue triangle ate joined to form the green triangle. The process of joining midpoints to form new triangles continues indefinitely
Are you doing geometric sequences? Fractals ? whatever ?
To find the total area of the infinite sequence of triangles, we can use the concept of geometric series.
Let's start by calculating the area of the largest red equilateral triangle. The formula to find the area of an equilateral triangle is A = (s^2 * √3) / 4, where s represents the length of the side.
Given that the side of the large red equilateral triangle is 8x units, the area can be calculated as follows:
A_red = (8x^2 * √3) / 4
Next, we consider the blue triangle formed by connecting the midpoints of the red triangle. Each side of the blue triangle is half the length of the corresponding side of the red triangle. Therefore, the side length of the blue triangle is 4x units.
Using the area formula for an equilateral triangle, the area of the blue triangle is:
A_blue = (4x^2 * √3) / 4
The same logic applies to the green triangle, where each side length is half the length of the corresponding side of the blue triangle. Therefore, the side length of the green triangle is 2x units.
Again, using the area formula for an equilateral triangle, the area of the green triangle is:
A_green = (2x^2 * √3) / 4
Notice that each subsequent triangle's area is reduced by a factor of four compared to the previous one because the side length is halved. This forms a geometric series:
A_red, A_blue, A_green, ... = A_red, A_red/4, A_red/16, ...
To find the total area of the infinite sequence of triangles, we need to sum up this series. The formula for the sum of an infinite geometric series is:
Sum = (a/1 - r)
Where 'a' is the first term of the series and 'r' is the common ratio.
In this case, the first term 'a' is A_red, and the common ratio 'r' is 1/4. Hence, the sum of the infinite geometric series is:
Sum = A_red / (1 - 1/4)
Simplifying the equation:
Sum = A_red / (3/4)
Therefore, the total area of the infinite sequence of triangles is:
Total Area = A_red * (4/3)
Substituting the expression for A_red:
Total Area = ((8x^2 * √3) / 4) * (4/3)
Simplifying further:
Total Area = 2x^2 * √3
Hence, the total area of the infinite sequence of triangles is 2x^2 * √3 units.