Can a row of five equilateral triangles tile a big equilateral triangle?

Solution 1:

I suppose I should post: I solved this on MathOverflow. The answer is YES: a size-45 triangle can be tiled.

I thank two insights from Josh B here: first that a rhombus with side length 15 can be tiled, and second the strategy to "select a different shape which does tile a triangle, then tile that shape with our $5$ triangle trapezoid."

This $15-15-15-30$ trapezoid can be tiled, and three such trapezoids can tile a triangle with side length $45$.

Solution 2:

Here's the minimal solution, a side-30 triangle. Also posted to MathOverflow here https://mathoverflow.net/questions/267095/can-a-row-of-five-equilateral-triangles-tile-a-big-equilateral-triangle.