Combine translations, reflections and turns, and explain tiling without gaps.
Free examples and worked answers. No account needed.
Practise the method.
Combine translations, reflections and turns, and explain tiling without gaps.
Combine translations, reflections and turns, and explain tiling without gaps.
Example 1
Two copies of any triangle can be joined by a 180∘ turn about the midpoint of one side to make a parallelogram. How can this help tile the plane?
Show worked answer
Answer: Translate copies of the parallelogram in rows to fill the plane.
A half-turn joins the pair of triangles into a parallelogram.
Repeated translations of that parallelogram fit opposite parallel edges together, creating a tessellation with no gaps or overlaps.
Example 2
One corner of a tile is at (1,2). Translate the tile 3 units right and 1 units up, then reflect it in the vertical y−axis. Where is that corner now? Give (x,y).
Tile corner before transformationsShow worked answer
Answer: (−4,3)
Translation gives (4,3).
Reflection in the y−axis reverses the horizontal coordinate and keeps the vertical coordinate: (−4,3).
Example 3
A tile corner starts at (3,4). Rotate it 90∘ clockwise about the origin, then translate it 2 units right. What are the final coordinates (x,y)?
Tile corner before turningShow worked answer
Answer: (6,−3)
A 90∘ clockwise rotation about the origin changes (x,y) to (y,−x), giving (4,−3).
Moving 2 right adds 2 to the first coordinate: (6,−3).
Put it into practice.
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