Are All Triangles Rep-4 Tiles?

Triangles, squares, rectangles, parallelograms, and rhombi have a fascinating property: they are all rep-4 tiles. This means that four copies of any of these shapes, when arranged properly, will form a larger version of the original shape.

The Definition of a Rep-4 Tile

A rep-4 tile is a geometric shape that can be divided into four congruent copies, each similar to the original shape. When these four copies are reassembled, they form a larger version of the original shape, with twice the length and height.

All triangles, squares, rectangles, parallelograms, and rhombi have this property, making them the simplest examples of rep-4 tiles.

Proving Triangles are Rep-4 Tiles

To prove that a triangle is a rep-4 tile, simply divide it into four congruent triangles by connecting the midpoints of its sides. These four triangles will be similar to the original, and when reassembled, they will form a larger triangle.

This property holds true for any triangle, regardless of its size or angles. The only requirement is that the four copies must be arranged in a specific way to form the larger triangle.

It’s important to note that while all triangles are rep-4 tiles, not all rep-4 tiles are triangles. Other shapes, such as squares, rectangles, parallelograms, and rhombi, also possess this property.