NCERT Solutions & Concepts for Class 7 Maths Chapter 14 Symmetry Exercise 14.3

Understanding Line and Rotational Symmetry Together

Understanding the connection between line and rotational symmetry is the key to mastering NCERT Exercise 14.3. Line symmetry is a static property where a line of symmetry divides a figure into two identical halves that act as mirror images of each other. Rotational symmetry, on the other hand, is a dynamic property: as a figure rotates around a fixed point called the 'center of rotation', it fits onto itself one or more times during a full 360-degree turn.

For example, consider a square. It has 4 lines of symmetry. When rotated around its center point (where its diagonals intersect), it looks exactly the same at 90 degrees, 180 degrees, 270 degrees, and 360 degrees. Therefore, its order of rotational symmetry is 4, and its angle of rotation is 360° / 4 = 90°. For regular polygons, the number of lines of symmetry is always equal to the order of rotational symmetry. However, this is not true for all shapes. A parallelogram has rotational symmetry of order 2 but has zero lines of symmetry. Recognizing these differences is exactly what Exercise 14.3 tests you on.

Important Symmetry Terminologies

Center of Rotation

The fixed point about which a 2D shape is rotated. For highly symmetrical shapes, this is the geometric center (e.g., the intersection point of the diagonals in a rectangle).

Order of Rotational Symmetry

The total number of times a shape looks identical to its original position during one full 360-degree rotation.

Angle of Rotation

The minimum angle of turn required for a shape to look exactly as it did in its starting position. It is calculated as 360° divided by the Order of Rotational Symmetry.

Exam Tips & Common Student Pitfalls

Here are crucial tips directly from YoLearn AI to help you score 100% on symmetry questions:

1. The Divisibility Rule for Angles: For any angle to be a valid angle of rotational symmetry, it must divide 360° perfectly without leaving a remainder. For instance, 45° is a valid angle of rotation ($360 / 45 = 8$), but 17° is not ($360 / 17 \approx 21.17$).

1. The Rectangle Trap: Many students lose marks by drawing diagonals as lines of symmetry for a rectangle. A rectangle has only 2 lines of symmetry (joining the midpoints of opposite sides), even though its rotational symmetry order is also 2.

1. Order 1 Meaning: Every object looks identical after a full 360° turn. Therefore, we say an object has rotational symmetry ONLY if its order of rotation is greater than 1 (i.e., it looks identical at an angle less than 360°).

Practice Questions with Solutions

• Q: Name any two figures that have both line symmetry and rotational symmetry of order more than 1. A: Step 1: Identify figures that have line symmetry (can be folded into matching halves) and also look identical after a rotation of less than 360 degrees. Step 2: Let's consider a Circle. A circle has infinite lines of symmetry and can be rotated by any angle around its center to look identical. Hence, its order of rotational symmetry is infinite. Step 3: Let's consider an Equilateral Triangle. It has 3 lines of symmetry. It also has rotational symmetry of order 3 because it looks identical at 120 degrees, 240 degrees, and 360 degrees. Final answer: An equilateral triangle and a circle are two perfect examples of figures having both line symmetry and rotational symmetry of order more than 1.
• Q: Draw, wherever possible, a rough sketch of a triangle with both line and rotational symmetry of order more than 1. A: Step 1: Recall the types of triangles: Scalene, Isosceles, and Equilateral. Step 2: Check an Isosceles Triangle. It has 1 line of symmetry, but its rotational symmetry order is only 1 (it must be rotated a full 360 degrees to look the same). Step 3: Check an Equilateral Triangle. It has 3 lines of symmetry (the three medians). Let's test its rotational symmetry. The angle of rotation is 360° / 3 = 120°. Since the order (3) is greater than 1, this triangle fits all criteria. Final answer: An Equilateral Triangle is the required triangle. To draw it, sketch a triangle with all three sides equal and draw three dashed lines from each vertex to the opposite side's midpoint to represent the lines of symmetry.
• Q: If a figure has two or more lines of symmetry, must it have rotational symmetry of order more than 1? A: Step 1: Let us test this hypothesis with known shapes. Step 2: Take a Rectangle. It has 2 lines of symmetry. If we rotate a rectangle by 180 degrees around the point of intersection of its diagonals, it looks identical. Its rotational symmetry order is 2, which is more than 1. Step 3: Take a Square. It has 4 lines of symmetry, and its rotational symmetry order is 4. Step 4: Geometrically, if a shape has 'n' lines of symmetry intersecting at a central point, that point acts as a center of rotation, ensuring a rotational symmetry of order 'n'. Since n is 2 or more, the rotational order will always be more than 1. Final answer: Yes, if a figure has two or more lines of symmetry, it must have rotational symmetry of order more than 1.
• Q: Can we have a rotational symmetry of order more than 1 whose angle of rotation is (i) 45°? (ii) 17°? A: Step 1: Use the mathematical relationship: $\text{Order of Rotational Symmetry} = 360^\circ / \text{Angle of Rotation}$. For the order to be valid, it must be a whole number greater than 1. Step 2: For (i) 45°: $\text{Order} = 360^\circ / 45^\circ = 8$. Since 8 is a whole number greater than 1, a shape (like a regular octagon) can have an angle of rotation of 45°. Step 3: For (ii) 17°: $\text{Order} = 360^\circ / 17^\circ \approx 21.17$. Since 21.17 is not an integer, we cannot have a shape with an angle of rotation of 17°. Final answer: (i) Yes, because 360 is perfectly divisible by 45. (ii) No, because 360 is not divisible by 17.

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