Determine if the given regular figures form semiregular tessellations.
#12) Triangle, decagon, 15-gon
To determine if the given regular figures form semiregular tessellations, we need to check if each regular figure fits together to completely cover a flat surface in a repeating pattern without any gaps or overlaps.
A semiregular tessellation is formed when at least two different regular polygons are used, and each vertex in the tessellation is surrounded by the same arrangement of polygons.
In this case, we have a triangle, a decagon, and a 15-gon. Let's analyze each regular figure individually:
1) Triangle: A triangle can form a tessellation by itself, as it can fully cover a plane when arranged properly. Since it is already included in the given figures, we don't need to consider it further.
2) Decagon: A decagon cannot tessellate by itself, as the internal angles of a regular decagon (with equal sides) do not allow for a repeating pattern without gaps or overlaps.
3) 15-gon: A 15-gon, also known as a pentadecagon, cannot tessellate by itself for the same reason as the decagon. The internal angles of a regular 15-gon do not allow for a repeating pattern that covers a plane without gaps or overlaps.
Based on this analysis, neither the decagon nor the 15-gon can form tessellations by themselves. Therefore, the given regular figures (triangle, decagon, and 15-gon) do not form semiregular tessellations together.