KS3 and KS4 Transformations Worksheets - Transformations of Graphs ... - Free Printable

Problem Overview:

The task involves analyzing the rotational symmetry and reflective symmetry of various polygons. Here's a breakdown of what needs to be done:

1. Rotational Symmetry: Determine the order of rotational symmetry for each shape. The order of rotational symmetry is the number of times a shape looks identical as it is rotated through 360°.
2. Reflective Symmetry: Identify and draw all lines of reflective symmetry for each shape. Reflective symmetry occurs when a shape can be divided into two identical halves by one or more lines.

Solution:

#### Section A: Analyzing Each Shape

1. Square
- Rotational Symmetry: A square looks identical after rotations of 90°, 180°, 270°, and 360°. Therefore, the order of rotational symmetry is 4.
- Reflective Symmetry: A square has 4 lines of reflective symmetry: 2 diagonals and 2 lines passing through the midpoints of opposite sides.
- Draw these lines as shown below:
```
+---+
| / |
|/ |
+---+
```

2. Kite
- Rotational Symmetry: A kite only looks identical after a rotation of 360°. Therefore, the order of rotational symmetry is 1.
- Reflective Symmetry: A kite has 1 line of reflective symmetry: the vertical line passing through the longer diagonal.
- Draw this line as shown below:
```
/\
/ \
/____\
```

3. Trapezium
- Rotational Symmetry: A trapezium (non-isosceles) only looks identical after a rotation of 360°. Therefore, the order of rotational symmetry is 1.
- Reflective Symmetry: A general trapezium does not have any lines of reflective symmetry unless it is an isosceles trapezium. For a non-isosceles trapezium, there are no lines of reflective symmetry.
- Draw no lines if it is a general trapezium.

4. Rhombus
- Rotational Symmetry: A rhombus looks identical after rotations of 180° and 360°. Therefore, the order of rotational symmetry is 2.
- Reflective Symmetry: A rhombus has 2 lines of reflective symmetry: the two diagonals.
- Draw these lines as shown below:
```
+---+
|\ /|
| X |
|/ \|
+---+
```

5. Rectangle
- Rotational Symmetry: A rectangle looks identical after rotations of 180° and 360°. Therefore, the order of rotational symmetry is 2.
- Reflective Symmetry: A rectangle has 2 lines of reflective symmetry: one horizontal and one vertical, passing through the midpoints of opposite sides.
- Draw these lines as shown below:
```
+---+
| |
+---+
```

6. Parallelogram
- Rotational Symmetry: A parallelogram (non-rectangular, non-rhombic) only looks identical after a rotation of 360°. Therefore, the order of rotational symmetry is 1.
- Reflective Symmetry: A general parallelogram does not have any lines of reflective symmetry unless it is a rectangle or rhombus. For a non-special parallelogram, there are no lines of reflective symmetry.
- Draw no lines if it is a general parallelogram.

7. Arrow Head
- Rotational Symmetry: An arrow head only looks identical after a rotation of 360°. Therefore, the order of rotational symmetry is 1.
- Reflective Symmetry: An arrow head has 1 line of reflective symmetry: the vertical line passing through the apex.
- Draw this line as shown below:
```
/\
/ \
/____\
```

8. Equilateral Triangle
- Rotational Symmetry: An equilateral triangle looks identical after rotations of 120°, 240°, and 360°. Therefore, the order of rotational symmetry is 3.
- Reflective Symmetry: An equilateral triangle has 3 lines of reflective symmetry: each line passes through a vertex and the midpoint of the opposite side.
- Draw these lines as shown below:
```
/\
/__\
```

9. Regular Pentagon
- Rotational Symmetry: A regular pentagon looks identical after rotations of 72°, 144°, 216°, 288°, and 360°. Therefore, the order of rotational symmetry is 5.
- Reflective Symmetry: A regular pentagon has 5 lines of reflective symmetry: each line passes through a vertex and the midpoint of the opposite side.
- Draw these lines as shown below:
```
*---*
/ \
/ \
* *
\ /
\ /
*---*
```

10. Regular Hexagon
- Rotational Symmetry: A regular hexagon looks identical after rotations of 60°, 120°, 180°, 240°, 300°, and 360°. Therefore, the order of rotational symmetry is 6.
- Reflective Symmetry: A regular hexagon has 6 lines of reflective symmetry: 3 lines pass through opposite vertices, and 3 lines pass through the midpoints of opposite sides.
- Draw these lines as shown below:
```
*---*---*
| \ / |
| \ / |
| X |
| / \ |
| / \ |
*---*---*
```

11. Regular Octagon
- Rotational Symmetry: A regular octagon looks identical after rotations of 45°, 90°, 135°, 180°, 225°, 270°, 315°, and 360°. Therefore, the order of rotational symmetry is 8.
- Reflective Symmetry: A regular octagon has 8 lines of reflective symmetry: 4 lines pass through opposite vertices, and 4 lines pass through the midpoints of opposite sides.
- Draw these lines as shown below:
```
*---*---*---*
| \ / \ / |
| \ / \ / |
| X X |
| / \ / \ |
| / \ / \ |
*---*---*---*
```

12. Regular Nonagon
- Rotational Symmetry: A regular nonagon looks identical after rotations of 40°, 80°, 120°, 160°, 200°, 240°, 280°, 320°, and 360°. Therefore, the order of rotational symmetry is 9.
- Reflective Symmetry: A regular nonagon has 9 lines of reflective symmetry: each line passes through a vertex and the midpoint of the opposite side.
- Draw these lines as shown below:
```
*---*---*---*
| \ / \ / |
| \ / \ / |
| X X |
| / \ / \ |
| / \ / \ |
*---*---*---*
```

Final Answer:

Summarizing the results in a table format:

| Shape | Rotational Symmetry Order | Lines of Reflective Symmetry |
|------------------|---------------------------|-------------------------------|
| Square | 4 | 4 |
| Kite | 1 | 1 |
| Trapezium | 1 | 0 (general case) |
| Rhombus | 2 | 2 |
| Rectangle | 2 | 2 |
| Parallelogram | 1 | 0 (general case) |
| Arrow Head | 1 | 1 |
| Equilateral Triangle | 3 | 3 |
| Regular Pentagon | 5 | 5 |
| Regular Hexagon | 6 | 6 |
| Regular Octagon | 8 | 8 |
| Regular Nonagon | 9 | 9 |

$$
\boxed{
\begin{array}{|c|c|c|}
\hline
\text{Shape} & \text{Rotational Symmetry Order} & \text{Lines of Reflective Symmetry} \\
\hline
\text{Square} & 4 & 4 \\
\text{Kite} & 1 & 1 \\
\text{Trapezium} & 1 & 0 \text{ (general case)} \\
\text{Rhombus} & 2 & 2 \\
\text{Rectangle} & 2 & 2 \\
\text{Parallelogram} & 1 & 0 \text{ (general case)} \\
\text{Arrow Head} & 1 & 1 \\
\text{Equilateral Triangle} & 3 & 3 \\
\text{Regular Pentagon} & 5 & 5 \\
\text{Regular Hexagon} & 6 & 6 \\
\text{Regular Octagon} & 8 & 8 \\
\text{Regular Nonagon} & 9 & 9 \\
\hline
\end{array}
}
$$