The equilateral triangle with 4 cm side is decreasing infinitely by 1/2. Determine the area of the triangles.
looks like a G.P. with
a = 4√3
r = 1/4
S = a/(1-r) = 16/√3
To determine the area of the infinitely decreasing equilateral triangles, we can use the concept of geometric series.
First, let's find the area of the original equilateral triangle with a side length of 4 cm. The formula to find the area of an equilateral triangle is:
Area = (√3 / 4) * side^2
Therefore, the area of the original equilateral triangle is:
Area = (√3 /4) * (4 cm)^2
Area = (√3 / 4) * 16 cm^2
Area = (√3 / 4) * 16 cm^2
Area = (√3 / 4) * 16 cm^2
Area = √3 * 4 cm^2
Area = 4√3 cm^2
Now, since the triangle is decreasing infinitely by 1/2, each subsequent triangle will have a side length that is half of the previous triangle. This forms a geometric progression with a common ratio of 1/2.
Using the formula for the sum of an infinite geometric series, we can find the total area of all the infinitely decreasing triangles:
Sum = a / (1 - r)
In this case, the first term (a) is 4√3 cm^2 and the common ratio (r) is 1/2.
Sum = (4√3 cm^2) / (1 - 1/2)
Sum = (4√3 cm^2) / (1/2)
Sum = 8√3 cm^2 * 2
Sum = 16√3 cm^2
Therefore, the total area of all the infinitely decreasing equilateral triangles is 16√3 cm^2.