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Shapes and Symmetry Exploration

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Shapes and Symmetry Exploration

The school canteen is planning a new geometric floor pattern. Solve each shape and symmetry challenge to help Hana and Nikau create a design that fits together neatly. Show working where needed.

Shape detectives

Use the properties of polygons to identify and classify shapes.

1.Hana sketches a four-sided shape with four right angles. Its opposite sides are equal in length, but not all four sides are equal. What is the most specific name for her shape?

2.Nikau draws a polygon with six sides. What is this polygon called, and how many interior angles does it have?

3.Which statement must be true for every square?

It has four equal sides and four right angles.
It has exactly one pair of parallel sides.
Its opposite sides are different lengths.
It has no lines of symmetry.

4.A floor tile is a quadrilateral with exactly one pair of parallel sides. What is this quadrilateral called in New Zealand mathematics?

Symmetry and transformations

A line of symmetry is a mirror line: reflecting a shape across it leaves the shape unchanged.

5.A regular hexagonal tile has six equal sides and six equal angles. How many lines of symmetry does it have?

6.Mere reflects a triangular motif across a vertical mirror line. The original motif has a vertex 4 cm to the left of the line and 2 cm above it. Where is the matching vertex in the reflected motif?

7.A square tile is turned a quarter-turn (90°) about its centre. Does it look unchanged after the turn? Explain using rotational symmetry.

8.Complete the sentence: A shape has line symmetry when a reflection across a line maps the shape .

Build a canteen floor pattern

Use your geometry knowledge to plan a repeating design. A tessellation covers a flat surface with no gaps or overlaps.

9.Which regular polygon can tessellate a floor by itself because its interior angle fits an exact number of times around a point?

Regular pentagon, interior angle 108°
Equilateral triangle, interior angle 60°
Regular octagon, interior angle 135°
Regular decagon, interior angle 144°

10.A regular hexagon has an interior angle of 120°. How many regular hexagon corners can meet at one point without leaving a gap or overlapping?

11.Nikau says, “Every shape with line symmetry can tessellate by itself.” Is he correct? Give one counterexample and explain briefly.

12.Design a small repeating tile motif for the canteen floor. Draw a polygon, mark at least one line of symmetry, and show how two copies meet along an edge. Label the polygon and state how many lines of symmetry your motif has.