GSEB Class 7 Maths Solutions Chapter 14 Symmetry Exercise 14.3

Gujarat Board Textbook Solutions Class 7 Maths Chapter 14 Symmetry Ex 14.3

Question 1. Name any two figures that have both line symmetry and rotational symmetry.
Answer: An equilateral triangle and a circle both possess line symmetry and rotational symmetry.
In simple words: Two shapes that have both kinds of symmetry are an equilateral triangle and a circle.

Exam Tip: Recall the definitions of both line and rotational symmetry to identify suitable figures. Common examples include regular polygons and circles.

Question 2. Draw, wherever possible, a rough sketch of
(i) a triangle with both line and rotational symmetries of order more than 1.
(ii) a triangle with only line symmetry and no rotational symmetry of order more than 1.
(iii) a quadrilateral with a rotational symmetry of order more than 1 but not a line symmetry,
(iv) a quadrilateral with line symmetry but not a rotational symmetry of order more than 1.
Answer:
(i) An equilateral triangle possesses three lines of symmetry. It also exhibits rotational symmetry of order three.
(ii) Such a triangle cannot exist or be drawn.
(iii) A parallelogram (specifically one that is not a rhombus or a rectangle) fits this description, as shown below.
(iv) It is not feasible to draw such a quadrilateral.
In simple words: For (i), an equilateral triangle works. For (ii) and (iv), you cannot draw such shapes. For (iii), a parallelogram (not a special one like a square) is what you would draw.

Exam Tip: Remember the specific properties of different geometric shapes. An equilateral triangle is a regular polygon and thus has both symmetries. Parallelograms (non-rhombus/rectangle) have rotational symmetry but not line symmetry.

Question 3. If a figure has two or more lines of symmetry, should it have rotational symmetry of order more than 1?
Answer: Yes, it should indeed possess rotational symmetry of order more than one.
In simple words: If a shape has at least two lines of symmetry, it will also spin to look the same more than once.

Exam Tip: Figures with multiple lines of symmetry are often regular polygons or other symmetrical shapes that inherently possess rotational symmetry.

Question 4. Fill in the blanks:
Answer:

Exam Tip: Memorize the rotational properties of common geometric shapes. For regular polygons, the order of rotation equals the number of sides, and the angle of rotation is \( 360^\circ \) divided by the number of sides.

Question 5. Name the quadrilaterals which have both line and rotational symmetry of order more than 1.
Answer: The square, rectangle, and a rhombus are quadrilaterals that exhibit both line symmetry and rotational symmetry.
In simple words: Quadrilaterals like squares, rectangles, and rhombuses have both line symmetry (you can fold them) and rotational symmetry (you can spin them and they look the same).

Exam Tip: To score full marks, list all three common quadrilaterals that satisfy both conditions: square, rectangle, and rhombus. Explain briefly how each exhibits both types of symmetry.

Question 6. After rotating by \( 60^\circ \) about a centre, a figure looks exactly the same as its original position. At what other angles will this happen for the figure?
Answer: The figure will appear identical to its starting position at angles of \( 120^\circ \), \( 180^\circ \), \( 240^\circ \), \( 300^\circ \), and \( 360^\circ \).
In simple words: If a shape looks the same after turning it \( 60^\circ \), it will also look the same if you turn it by multiples of \( 60^\circ \) (like \( 120^\circ \), \( 180^\circ \), and so on) until a full circle.

Exam Tip: If a figure has rotational symmetry at an angle \( x \), it will also have rotational symmetry at all multiples of \( x \) up to \( 360^\circ \).

Question 7. Can we have a rotational symmetry of order more than 1 whose angle of rotation is
(i) \( 45^\circ \)?
(ii) \( 17^\circ \)?
Answer:
(i) Yes, an angle of \( 45^\circ \) allows for rotational symmetry of order more than one.
(ii) No, an angle of \( 17^\circ \) does not allow for rotational symmetry of order more than one.
In simple words: For a shape to have rotational symmetry, the full circle of \( 360^\circ \) must be perfectly divided by the rotation angle. \( 45^\circ \) divides \( 360^\circ \) evenly (8 times), but \( 17^\circ \) does not.

Exam Tip: A figure has rotational symmetry of order greater than 1 with an angle of rotation \( x \) if and only if \( 360^\circ \) is perfectly divisible by \( x \). Calculate \( 360^\circ / x \) to determine if it results in a whole number.