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
Urgent##1. Solve for x if the sum of all the areas is given by 19200 square root 3
If you draw it you can see that
An = An-1 * .25
or
a , .25 a , .25^2 a etc
we have a sum of a geometric series here with r = .25
see
http://www.mathsisfun.com/algebra/sequences-sums-geometric.html
the first term a is area of triangle with 8 x sides
a = 2 * 4x * 4x sqrt 3
= 32 sqrt 3
sun of infinite geometric series
= a /(1-r)
= 32 sqrt 3 /.75
= 42.67 sqrt 3
so A total = 32 x^2 sqrt 3/.75
so
32 x^2 sqrt 3/.75 = 19200 sqrt 3
x^2 = 450
x = 21.2
check my arithmetic
To solve for x, we first need to understand the properties of the triangles formed.
Let's start by marking the midpoints of the sides of the red triangle. This will form a smaller blue triangle within the red triangle. The blue triangle is also an equilateral triangle since its sides are formed by connecting the midpoints of the larger triangle.
The ratio of the sides of the large red triangle to the small blue triangle is 2:1. This means that the side length of the blue triangle is (1/2) * 8x = 4x units.
Similarly, we can mark the midpoints of the sides of the blue triangle to form a smaller green triangle. Again, the ratio of the sides of the blue triangle to the green triangle is 2:1. Therefore, the side length of the green triangle is (1/2) * 4x = 2x units.
We can continue this process to form even smaller triangles, with each subsequent triangle having sides half the length of the previous one.
Now, let's calculate the sum of the areas of these triangles. Since the area of an equilateral triangle is given by A = (sqrt(3)/4) * s^2, where s is the side length, we can determine the area of each triangle.
The area of the red triangle = (sqrt(3)/4) * (8x)^2 = 16sqrt(3) * x^2,
The area of the blue triangle = (sqrt(3)/4) * (4x)^2 = 4sqrt(3) * x^2,
The area of the green triangle = (sqrt(3)/4) * (2x)^2 = sqrt(3) * x^2,
Looking at this pattern, we notice that the areas form a geometric sequence where each term is one-fourth the previous term.
Now, let's calculate the sum of an infinite geometric sequence using the sum formula.
The sum of an infinite geometric series is given by S = a / (1 - r), where 'a' is the first term and 'r' is the common ratio.
In this case, the first term 'a' is 16sqrt(3) * x^2, and the common ratio 'r' is 1/4.
Using S = 19200sqrt(3), we have:
19200sqrt(3) = (16sqrt(3) * x^2) / (1 - 1/4),
19200 = 16x^2 / (3/4),
19200 = 64x^2 / 3,
Now, let's solve for x:
Multiply both sides of the equation by (3/64):
19200 * (3/64) = 64x^2,
900 = x^2,
Take the square root of both sides:
√900 = √x^2,
30 = x.
Therefore, the value of x is 30 units.