Symmetry Ex 14.2 Class 7 NCERT Maths: Mastering Rotational Symmetry

What is Rotational Symmetry?

Rotational symmetry occurs when a figure looks exactly the same after it has been rotated by some amount less than a full 360 degrees around a fixed point. This fixed point is called the centre of rotation. Think of a fan: when it spins, its blades appear to be in the same position multiple times before completing a full circle. That's rotational symmetry in action!

The order of rotational symmetry is the number of times a figure looks identical to its original position during a complete 360-degree rotation. Importantly, the original position (after a 360-degree turn) is always counted as one position. If a figure only looks the same after a full 360-degree rotation, it has an order of rotational symmetry of 1. The smallest angle by which a figure must be rotated to look identical to its original position is called the angle of rotational symmetry. We can find this angle using the formula: Angle of Rotational Symmetry = 360° / Order of Rotational Symmetry.

For example, a square looks exactly the same after rotating by 90°, 180°, 270°, and 360°. So, it aligns with its original position 4 times, giving it an order of rotational symmetry of 4. Its angle of rotational symmetry would be 360° / 4 = 90°.

Finding the Order of Rotational Symmetry: Step-by-Step

1. Step 1: Identify the Centre of Rotation — Locate the central point around which the figure will rotate. For many regular shapes, this is often the geometric centre.
2. Step 2: Imagine or Trace the Figure — Mentally picture the figure rotating, or if possible, trace the figure on a piece of paper and place it over the original figure, using a pin at the centre of rotation.
3. Step 3: Rotate the Figure Gradually — Slowly rotate the traced figure (or imagine rotating it) around the centre of rotation. Keep track of the rotation.
4. Step 4: Count Identical Positions — Count how many times the rotated figure perfectly coincides with its original position before completing a full 360° turn. Remember to include the initial position (0° or 360° rotation) in your count. This count is the order of rotational symmetry.

Worked Examples: Mastering Rotational Symmetry

• Example 1: Square 1. Centre of Rotation: The intersection of its diagonals. 2. Rotation: Rotate the square around its centre. 3. Identical Positions: It matches the original position at 90°, 180°, 270°, and 360° (original position). 4. Order: The square has an order of rotational symmetry of 4. Angle of rotation = 360° / 4 = 90°.
• Example 2: Equilateral Triangle 1. Centre of Rotation: The intersection of its medians (centroid). 2. Rotation: Rotate the equilateral triangle around its centre. 3. Identical Positions: It matches the original position at 120°, 240°, and 360°. 4. Order: An equilateral triangle has an order of rotational symmetry of 3. Angle of rotation = 360° / 3 = 120°.
• Example 3: Rectangle (not a square) 1. Centre of Rotation: The intersection of its diagonals. 2. Rotation: Rotate the rectangle around its centre. 3. Identical Positions: It matches the original position at 180° and 360°. 4. Order: A rectangle has an order of rotational symmetry of 2. Angle of rotation = 360° / 2 = 180°.
• Example 4: Circle 1. Centre of Rotation: Its geometric centre. 2. Rotation: Rotate the circle around its centre. 3. Identical Positions: A circle looks the same at every single angle of rotation. 4. Order: A circle has an infinite order of rotational symmetry.

Exam Tip: Avoiding Common Mistakes

When dealing with rotational symmetry, students often make a few common errors. Be careful not to confuse line symmetry with rotational symmetry; a shape can have one, both, or neither. For instance, an isosceles triangle has line symmetry but only an order of 1 for rotational symmetry (unless it's equilateral). Always remember to include the original position (after a 360° rotation) when counting the order of rotational symmetry. If a shape looks identical at 0° (its starting position), then again at 180°, and finally at 360°, its order is 2, not 1. Lastly, ensure you correctly identify the centre of rotation; for complex figures, this might not always be immediately obvious. Practice drawing and rotating shapes to build your visual understanding.

Practice Questions with Solutions

• Q: What is the order of rotational symmetry for a regular hexagon? A: Step 1: Identify the centre of rotation as the centre of the hexagon. Step 2: A regular hexagon has 6 equal sides and 6 equal angles. Step 3: When rotated, it will look identical at 6 different positions during a 360° turn (at 60°, 120°, 180°, 240°, 300°, and 360°). Final answer: The order of rotational symmetry for a regular hexagon is 6.
• Q: Determine the order of rotational symmetry for the letter 'S'. A: Step 1: Identify the approximate centre of the letter 'S'. Step 2: Rotate the letter 'S' around its centre. Step 3: It will look exactly the same after a 180° rotation and again at 360° (its original position). Final answer: The order of rotational symmetry for the letter 'S' is 2.
• Q: Find the order of rotational symmetry for an isosceles triangle (which is not equilateral). A: Step 1: Identify the centre of rotation. Step 2: Rotate the isosceles triangle. Step 3: An isosceles triangle (with only two equal sides) will only look identical to its original position after a full 360° rotation. Final answer: The order of rotational symmetry for an isosceles triangle is 1.
• Q: What is the order of rotational symmetry for a parallelogram? A: Step 1: The centre of rotation for a parallelogram is the intersection of its diagonals. Step 2: Rotate the parallelogram around this centre. Step 3: A parallelogram will look identical to its original position after a 180° rotation and again at 360°. Final answer: The order of rotational symmetry for a parallelogram is 2.

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