RS Aggarwal Class 7 Mathematics Solutions Chapter 18 Reflection and Rotational Symmetry

Question 1. Which shape has no line of symmetry?
Answer: A shape with no line of symmetry is one where you cannot fold it along any line and have both halves match perfectly.

Exam Tip: Look for irregular shapes that don't have matching halves when folded in any direction.

Question 2. A line joining the midpoints of its opposite sides is a line of symmetry for which shape?
Answer: A rectangle has a line of symmetry that connects the midpoints of its opposite sides. When you fold a rectangle along this line, both halves overlap completely.

Exam Tip: Remember that a rectangle has exactly two such lines of symmetry - one horizontal and one vertical.

Question 3. How many lines of symmetry does a square have?
Answer: A square possesses four lines of symmetry. These include both of its diagonals and both lines that connect the midpoints of its opposite sides.

Exam Tip: Always draw the figure and mark all possible fold lines to verify the count of symmetry lines.

Question 4. A diamond shape is symmetrical about which lines?
Answer: A diamond is symmetrical about each of its diagonals. The diagonals act as mirror lines, reflecting one half of the shape onto the other.

Exam Tip: For quadrilaterals, always check both diagonals as potential lines of symmetry.

Question 5. How many lines of symmetry does a rhombus have?
Answer: A rhombus has two lines of symmetry, which are its two diagonals. When folded along either diagonal, the two halves fit together perfectly.

Exam Tip: Do not confuse a rhombus with a square - while both have two diagonal lines of symmetry, only a square has four total lines.

Question 6. How many lines of symmetry does a circle have?
Answer: A circle is symmetrical about every single one of its diameters. Since a circle has an infinite number of diameters, it possesses an unlimited number of lines of symmetry. Therefore, a circle has unlimited number of lines of symmetry.

Exam Tip: A circle is the only common geometric shape with infinite symmetry lines - use this as a key identifying feature.

Question 7. What is the line of symmetry for an isosceles triangle?
Answer: An isosceles triangle has just one line of symmetry. This line runs from the vertex connecting the two equal sides down to the midpoint of the base. The altitude from the vertex angle to the base serves as this symmetry line, but the triangle does not have any rotational symmetry.

Exam Tip: In an isosceles triangle, the line of symmetry is always the altitude from the vertex between the two equal sides.

Question 8. What is the line of symmetry for a kite-shaped figure?
Answer: The line of symmetry for a kite is one of its diagonals - specifically, the diagonal that connects the two vertices where unequal sides meet. This diagonal divides the kite into two congruent halves. AC serves as the line of symmetry, dividing the shape into two equal parts.

Exam Tip: A kite has only one line of symmetry, not two - remember which diagonal is the axis of symmetry.

Question 9. How many lines of symmetry does the letter O have?
Answer: The letter O in the English alphabet has two lines of symmetry. It is symmetrical about both its horizontal line and its vertical line. These two perpendicular lines pass through the centre of the letter, dividing it into matching halves.

Exam Tip: The letter O, like a circle, has multiple lines of symmetry passing through its centre.

Question 10. Draw lines of symmetry for the following figures.
(i) A right-angled triangle
(ii) An arrow pointing to the right
(iii) An arrow pointing upward
(iv) A cross or plus sign
Answer: Lines of symmetry are shown by the dotted lines.
(i) A right-angled triangle has one line of symmetry if it is also isosceles, running from the right angle to the midpoint of the hypotenuse.
(ii) An arrow pointing right has one line of symmetry - a horizontal line through its middle.
(iii) An arrow pointing upward has one line of symmetry - a vertical line running down its centre.
(iv) A cross or plus sign has four lines of symmetry - two diagonals and two straight lines (one vertical and one horizontal) through its centre.

Exam Tip: Always fold mentally along each potential line to verify symmetry, and mark only the correct ones in your answer.

Question 11. State whether each of the following statements is true or false.
(i) Every figure has at least one line of symmetry.
(ii) A triangle always has at least one line of symmetry.
(iii) An equilateral triangle has three lines of symmetry.
(iv) A rhombus has four lines of symmetry.
(v) A square has four lines of symmetry.
(vi) A rectangle has four lines of symmetry.
(vii) Each of the letters H, I, O and X has two lines of symmetry.
(viii) No figure can have both lines of symmetry and rotational symmetry.
Answer:
(i) False - Many shapes, such as scalene triangles and irregular polygons, do not have any lines of symmetry.
(ii) False - Not all triangles have lines of symmetry. Scalene triangles (where all sides are different lengths) have no lines of symmetry.
(iii) True - An equilateral triangle has three lines of symmetry, one from each vertex to the midpoint of the opposite side.
(iv) False - A rhombus has only two lines of symmetry (its diagonals), not four.
(v) True - A square is symmetrical about both its diagonals and both lines joining the midpoints of its opposite sides. So, a square has four lines of symmetry.
(vi) False - A rectangle is symmetrical about only the two lines joining the midpoints of its opposite sides. So, a rectangle has two lines of symmetry, not four.
(vii) True - Each of the letters H, I, O and X in the English alphabet system is symmetrical about its horizontal and vertical line, in the middle of the letters. So, all these letters have two lines of symmetry.
(viii) False - Many shapes can possess both. For example, a square has four lines of symmetry and also has rotational symmetry of order 4.

Exam Tip: For each statement, draw or visualize the figure and test all possible fold lines before deciding true or false.

Exercise 18(B)

Question 1. How many lines of symmetry does an equilateral triangle have, and what is its order of rotational symmetry?
Answer: An equilateral triangle has three lines of symmetry. Each line runs from one vertex to the midpoint of the opposite side. When a figure can be rotated to a position that looks exactly the same without any visible change, the count of such positions is called its order of rotational symmetry. For an equilateral triangle, the order of rotational symmetry is 3.

Exam Tip: The number of lines of symmetry in an equilateral triangle equals its order of rotational symmetry - both are 3.

Question 2. If a rectangle is rotated by 180° or 360°, what does it look like?
Answer: If you rotate a rectangle by either 180° or 360°, it appears identical to its original position. When you flip it by 180°, it matches exactly as if no change occurred - it becomes symmetrical. This shows that when rotated through these angles, the rectangle remains unchanged in appearance.

Exam Tip: A 360° rotation always returns any shape to its original position, so check 180° to find the real rotational symmetry.

Question 3. If a square is rotated by 90°, 180°, 270° or 360°, what happens to its appearance?
Answer: When you rotate a square by 90°, 180°, 270° or 360°, the square looks exactly the same. It appears unchanged after each rotation. As a result, the order of rotational symmetry of a square is 4.

Exam Tip: The more angles at which a shape looks identical, the higher its order of rotational symmetry.

Question 4. How many lines of symmetry and what is the order of rotational symmetry for a rhombus?
Answer: A rhombus has 2 lines of symmetry. These are its two diagonals. When you flip the rhombus along either diagonal, both halves match perfectly. When you rotate the rhombus by either 180° or by 360°, it looks the same. Therefore, the rotational symmetry of a rhombus is 2.

Exam Tip: For quadrilaterals, always check both the number of fold lines and rotation angles separately - they often differ.

Question 5. Which letters have 2 lines of symmetry and an order of rotational symmetry of 2?
Answer: H, O and X are the three letters that have 2 lines of symmetry and their order of rotational symmetry is 2.

Exam Tip: Test each letter by rotating it 180° and folding it in all directions to confirm both properties.

Question 6. Does an isosceles triangle have a line of symmetry and does it have rotational symmetry?
Answer: The line of symmetry for an isosceles triangle is the angle bisector of its vertex angle. This line passes between the two equal sides and is positioned between them. Conversely, it does not have any rotational symmetry.

Exam Tip: An isosceles triangle always has exactly one line of symmetry - never confuse this with rotational properties.

Question 7. Does a trapezium have a line of symmetry?
Answer: Not every trapezium has a line of symmetry. Only an isosceles trapezium possesses a line of symmetry. In an isosceles trapezium, the line of symmetry is a vertical line running through the middle, perpendicular to the two parallel sides.

Exam Tip: Remember that "trapezium" in British English is "trapezoid" in American English - check which definition your textbook uses.

Question 8. Does a semicircle have a line of symmetry? Does it have rotational symmetry?
Answer: The perpendicular bisector of the diameter of a semicircle serves as its line of symmetry. No, a semicircle does not have any rotational symmetry as it fits itself only once during a complete rotation.

Exam Tip: A semicircle has exactly one line of symmetry - the perpendicular to its flat edge through the centre.

Question 9. Can a scalene triangle have both a line of symmetry and rotational symmetry?
Answer: A scalene triangle, where all three sides have different lengths, does not have either a line of symmetry or a rotational symmetry.

Exam Tip: Scalene triangles are the most asymmetrical of all triangles - they lack both types of symmetry.

Question 10. For the given figure, how many lines of symmetry does it have and what is its order of rotational symmetry?
Answer:
(i) The line of symmetry of the given figure is 1.
(ii) The order of rotational symmetry of the given figure is 0.

Exam Tip: When a figure has order of rotational symmetry as 0, it means the figure looks the same only in its original position.

Question 11. For the given figure, how many lines of symmetry does it have and what is its order of rotational symmetry?
Answer:
(i) The given figure has 2 lines of symmetry.
(ii) The order of rotational symmetry of the given figure is 2.

Exam Tip: When a figure can be rotated to match itself, count how many times it repeats before returning to its start position.

Question 12. Give an example of a letter from the English alphabet that has no line of symmetry and has rotational symmetry of order 2.
Answer: The letter N is an example from the English alphabet system which has no line of symmetry and also has rotational symmetry of order 2.

Exam Tip: Look for letters that look the same when rotated 180° but don't have any fold lines - N is a classic example.