Symmetry and tessellationIB MYP Maths Extended: Subtopic test

1. 1
   A rhombus that is not a square.

   (a)

   How many lines of symmetry does the rhombus have?

   [1 mark]

   • A$4$
   • B$1$
   • C$2$
   • D$0$

   (b)

   What is the order of rotational symmetry of the rhombus?

   [1 mark]

   • A$2$
   • B$4$
   • C$1$
   • D$3$

   (c)

   Explain why the rhombus does not have rotational symmetry of order $4$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A tiler wants to cover a floor with tiles, leaving no gaps and no overlaps.

   (a)

   Which regular polygon cannot be used on its own as the tile?

   [1 mark]

   • AEquilateral triangle
   • BSquare
   • CRegular hexagon
   • DRegular pentagon

   (b)

   The tiler uses regular hexagons. How many hexagons meet at each vertex?

   [1 mark]

   • A$2$
   • B$3$
   • C$4$
   • D$6$

   (c)

   The tiler also considers using squares and regular octagons together. Show that one square and two regular octagons fit together at a vertex.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A quadrilateral tile has angles of $70^\circ$, $95^\circ$, $110^\circ$ and $x^\circ$. Many identical copies of the tile are available.

   (a)

   Find the value of $x$.

   [3 marks]

   (b)

   Explain why copies of this tile can be fitted together around a point with no gaps and no overlaps, and why the same method works for any quadrilateral.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A designer investigates which regular polygons tessellate on their own. The interior angle of a regular polygon with $n$ sides is $\left(180-\frac{360}{n}\right)^\circ$.

   (a)

   (i) Find the interior angle of a regular triangle, pentagon and hexagon.
   (ii) For each, find $360\div$ interior angle, and say which of the three tessellate on their own.
   (iii) Decide, with working, whether a regular octagon tessellates on its own.

   [6 marks]

   (b)

   The designer claims that no regular polygon with more than $6$ sides tessellates on its own. Justify the claim.

   [6 marks]

   Total for question 4: 12 marks

End of questions