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.

Question

1. Determines the lengths of the sides of the first three triangles.
2. State the heights of these triangles
3. What type of sequence is generated by the heights
4. Determine the areas of the triangles
5. Solve for x if the sum of all the areas is given by 19200 square root 3

Answer 1

since each triangle has sides 1/2 as big, the areas are 1/4 as big.

So, just form a G.P. of the desired quantities and work with that.