GSEB Class 7 Maths Solutions Chapter 14 સંમિતિ Exercise 14.3

Question 1. Tell us two digits that have both linear symmetry and rotational symmetry.
Answer: Let us look at the linear symmetry of the digit zero. Zero is circular, so it possesses numerous orders of linear symmetry. In the same way, zero also has an infinite order of rotational symmetry.
Now, let us examine the linear symmetry of the digit eight (8). The digit eight (8) has a degree of linear symmetry. Similarly, zero also has an infinite order of rotational symmetry. Both 0 and 8 possess two degrees of symmetry.
In simple words: The digits 0 and 8 both have linear symmetry (they can be folded in half to match) and rotational symmetry (they look the same after being turned). Zero is round, so it has endless lines of symmetry and can be spun any amount. The number 8 has two lines of symmetry and can be rotated 180 degrees.

Exam Tip: Remember that "linear symmetry" means you can fold it perfectly, and "rotational symmetry" means it looks the same after turning it a certain amount.

Question 2. For each of the following, draw a rough figure if possible:

Question 2. (i) A triangle that has both linear symmetry and rotational symmetry of more than one order.
Answer: An equilateral triangle has linear symmetry of more than one order. As shown in the figure, an equilateral triangle also possesses rotational symmetry of more than one order (it has 3).

In simple words: An equilateral triangle has many lines of symmetry and can be turned 3 times (120° each time) to look exactly the same.

Exam Tip: An equilateral triangle has 3 lines of symmetry and rotational symmetry of order 3, meaning it looks the same after rotations of 120°, 240°, and 360°.

Question 2. (ii) A triangle that has only one line of symmetry but no rotational symmetry.
Answer: An isosceles triangle is one that possesses only one line of symmetry but no rotational symmetry.

In simple words: An isosceles triangle can be folded in half along one line, but it won't look the same if you just turn it (unless you turn it a full 360 degrees).

Exam Tip: An isosceles triangle has two equal sides and two equal angles, which allows for only one line of symmetry.

Question 2. (iii) A quadrilateral that has rotational symmetry of more than one order but no linear symmetry.
Answer: A parallelogram is a quadrilateral that possesses rotational symmetry of more than one order but does not have linear symmetry.

In simple words: A parallelogram will look the same if you spin it around halfway (180 degrees), but you can't fold it in half perfectly like a square or rectangle.

Exam Tip: A parallelogram has point symmetry (rotational symmetry of order 2) around the intersection of its diagonals, but typically no line symmetry unless it's a rectangle or a rhombus.

Question 2. (iv) A quadrilateral that has only one line of symmetry but no rotational symmetry.
Answer: An isosceles trapezoid is a quadrilateral that possesses only one line of symmetry but no rotational symmetry.

In simple words: An isosceles trapezoid can be folded in half along a single line, but it will not look identical if you rotate it less than a full circle.

Exam Tip: An isosceles trapezoid has non-parallel sides of equal length, allowing for one line of symmetry that passes through the midpoints of the parallel sides.

Question 3. If a figure has two or more lines of symmetry, does it necessarily have rotational symmetry of order more than 1?
Answer: Yes, if a figure possesses two or more lines of symmetry, then its rotational symmetry order must be more than 1. For example, a square has 4 lines of symmetry and its rotational symmetry order is also 4.

In simple words: Yes, if a shape has two or more ways to fold it perfectly in half, it will also look the same if you turn it a certain amount (more than just a full circle). A square, for instance, has four folding lines and can be turned four ways to look the same.

Exam Tip: Most figures with multiple lines of symmetry (like regular polygons) also exhibit rotational symmetry, with the order of rotation often correlating with the number of lines of symmetry.

Question 4. Fill in the blanks:
Answer:

Exam Tip: Memorize the rotational properties of common geometric shapes. Understanding the relationship between the number of sides and the order/angle of rotational symmetry for regular polygons is key.

Question 5. Name a quadrilateral that has both linear symmetry and rotational symmetry of order more than 1.
Answer: Square, rectangle, and rhombus are such quadrilaterals that possess both linear symmetry and rotational symmetry of order more than 1.
In simple words: Shapes like squares, rectangles, and rhombuses are special because you can fold them perfectly in half and they also look the same after you turn them a bit.

Exam Tip: Recall the properties of various quadrilaterals; squares and rectangles are easy examples for both types of symmetry, while a rhombus has line symmetry along its diagonals and rotational symmetry of order 2.

Question 6. The figure looks like its original position after rotating 60° from the center. For what other angles will this happen?
Answer: If the figure appears like its original position after rotating 60° from the center, then the same figure will also look like its original position at 120°, 180°, 240°, 300°, and 360° angles.
In simple words: If a shape looks the same after turning 60 degrees, it will also look the same if you turn it by multiples of 60 degrees. So, 120, 180, 240, 300, and 360 degrees will also make it look unchanged.

Exam Tip: If a figure has rotational symmetry at a specific angle, it will also have symmetry at all multiples of that angle within 360 degrees. Divide 360 by the given angle to find the order of symmetry.

Question 7. For the angles given below, can we obtain rotational symmetry of order more than 1?

Question 7. (i) 45°
Answer: 360° can be completely divided by 45. Therefore, rotational symmetry of order more than 1 can be obtained for a 45° angle.
In simple words: Since 360 can be divided evenly by 45 (360 ÷ 45 = 8), a shape that rotates 45 degrees to look the same will have rotational symmetry of order more than one.

Exam Tip: To determine if an angle can result in rotational symmetry of order greater than 1, divide 360° by the given angle. If the result is a whole number greater than 1, then it's possible.

Question 7. (ii) 17°
Answer: 360° cannot be completely divided by 17°. Therefore, rotational symmetry of order more than 1 cannot be obtained for a 17° angle.
In simple words: Because 360 cannot be divided evenly by 17, a shape rotating by 17 degrees will not have rotational symmetry of an order higher than one.

Exam Tip: If 360° divided by the given angle does not result in a whole number, then the figure does not have rotational symmetry of that order, except for the trivial order 1 (360° rotation).