ML Aggarwal Class 7 Maths Solutions Chapter 14 Symmetry

Exercise 14(1)

Question 1. (i) Rotational symmetry of order 2
Answer: A figure has rotational symmetry of order 2 when it appears identical after being turned 180 degrees around its centre point.
In simple words: The shape looks the same if you spin it halfway around.

Exam Tip: Order 2 means the figure matches itself at exactly 180 degrees - this is the most common rotational symmetry in rectangles and parallelograms.

Question 1. (ii) Rotational symmetry of order 2
Answer: A figure has rotational symmetry of order 2 when it appears identical after being turned 180 degrees around its centre point.
In simple words: The shape looks the same if you spin it halfway around.

Exam Tip: Order 2 means the figure matches itself at exactly 180 degrees - this is the most common rotational symmetry in rectangles and parallelograms.

Question 1. (iii) No rotational symmetry
Answer: This figure does not have rotational symmetry. It does not look the same when rotated by any angle less than 360 degrees around its centre.
In simple words: No matter which direction you spin this shape, it never looks exactly like the original until you turn it all the way around.

Exam Tip: If a figure has no rotational symmetry, it has only an order of 1 (which means it matches only when rotated the full 360 degrees).

Question 1. (iv) Rotational symmetry of order 2
Answer: A figure has rotational symmetry of order 2 when it appears identical after being turned 180 degrees around its centre point.
In simple words: The shape looks the same if you spin it halfway around.

Exam Tip: Order 2 means the figure matches itself at exactly 180 degrees - this is the most common rotational symmetry in rectangles and parallelograms.

Question 1. (v) No rotational symmetry
Answer: This figure does not have rotational symmetry. It does not look the same when rotated by any angle less than 360 degrees around its centre.
In simple words: No matter which direction you spin this shape, it never looks exactly like the original until you turn it all the way around.

Exam Tip: If a figure has no rotational symmetry, it has only an order of 1 (which means it matches only when rotated the full 360 degrees).

Question 1. (vi) Rotational symmetry of order 4
Answer: A figure has rotational symmetry of order 4 when it appears identical after being turned 90 degrees around its centre point. This means it matches itself four times as you rotate it through 360 degrees.
In simple words: The shape looks the same when you spin it one quarter turn, half turn, three-quarter turn, and full turn.

Exam Tip: Order 4 is found in squares and some cross or star shapes - think of a shape that repeats itself four times as you rotate it.

Question 1. (vii) Rotational symmetry of order 4
Answer: A figure has rotational symmetry of order 4 when it appears identical after being turned 90 degrees around its centre point. This means it matches itself four times as you rotate it through 360 degrees.
In simple words: The shape looks the same when you spin it one quarter turn, half turn, three-quarter turn, and full turn.

Exam Tip: Order 4 is found in squares and some cross or star shapes - think of a shape that repeats itself four times as you rotate it.

Question 1. (viii) No rotational symmetry
Answer: This figure does not have rotational symmetry. It does not look the same when rotated by any angle less than 360 degrees around its centre.
In simple words: No matter which direction you spin this shape, it never looks exactly like the original until you turn it all the way around.

Exam Tip: If a figure has no rotational symmetry, it has only an order of 1 (which means it matches only when rotated the full 360 degrees).

Question 1. (ix) Rotational symmetry of order 2
Answer: A figure has rotational symmetry of order 2 when it appears identical after being turned 180 degrees around its centre point.
In simple words: The shape looks the same if you spin it halfway around.

Exam Tip: Order 2 means the figure matches itself at exactly 180 degrees - this is the most common rotational symmetry in rectangles and parallelograms.

Question 1. (x) Rotational symmetry of order 4
Answer: A figure has rotational symmetry of order 4 when it appears identical after being turned 90 degrees around its centre point. This means it matches itself four times as you rotate it through 360 degrees.
In simple words: The shape looks the same when you spin it one quarter turn, half turn, three-quarter turn, and full turn.

Exam Tip: Order 4 is found in squares and some cross or star shapes - think of a shape that repeats itself four times as you rotate it.

Question 1. (xi) Rotational symmetry of order 6
Answer: A figure has rotational symmetry of order 6 when it appears identical after being turned 60 degrees around its centre point. This means it matches itself six times as you rotate it through 360 degrees.
In simple words: The shape looks the same each time you turn it by 60 degrees - a perfect pattern that repeats six times going around.

Exam Tip: Order 6 is seen in regular hexagons and certain flower or star patterns - look for shapes that repeat naturally six times.

Question 1. (xii) Rotational symmetry of order 4
Answer: A figure has rotational symmetry of order 4 when it appears identical after being turned 90 degrees around its centre point. This means it matches itself four times as you rotate it through 360 degrees.
In simple words: The shape looks the same when you spin it one quarter turn, half turn, three-quarter turn, and full turn.

Exam Tip: Order 4 is found in squares and some cross or star shapes - think of a shape that repeats itself four times as you rotate it.

Question 2. (1) and (ii) have rotational symmetry of order greater than 1.
Answer: This statement is correct. Both figures from question 1 parts (i) and (ii) display rotational symmetry at an order above 1, meaning they match themselves when rotated at specific angles before completing a full 360-degree turn.
In simple words: Both of these shapes look the same when rotated by less than a full spin around.

Exam Tip: When you see "order greater than 1," it simply means the figure has some rotational symmetry - look for any angle that makes it match itself before the full rotation.

Question 3. Rhombus and equilateral triangle have both line of symmetry and rotational symmetry.
Answer: A rhombus has two lines of symmetry (its diagonals) and rotational symmetry of order 2 (it looks the same after a 180-degree turn). An equilateral triangle has three lines of symmetry and rotational symmetry of order 3 (it looks the same after a 120-degree turn). Both shapes combine line symmetry with rotational symmetry.
In simple words: Both shapes can be folded in half along certain lines, and both look the same when spun around.

Exam Tip: Remember that not all shapes with line symmetry have rotational symmetry - check both properties separately for each figure.

Question 4. Rectangle, Rhombus and Square have both line of symmetry and rotational symmetry of order more than 1.
Answer: A rectangle has two lines of symmetry and rotational symmetry of order 2. A rhombus has two lines of symmetry and rotational symmetry of order 2. A square has four lines of symmetry and rotational symmetry of order 4. All three shapes display both types of symmetry, and all have rotational order greater than 1.
In simple words: Each of these shapes can be folded along lines to match itself, and each spins to match itself at least twice before going all the way around.

Exam Tip: The square is the most symmetric of these three - it has the most lines of symmetry and the highest rotational order.

Question 5. (i) Rectangle
Answer: A rectangle has one horizontal line of symmetry running through its centre, dividing it into two identical halves.
In simple words: If you draw a line across the middle from left to right, each half folds onto the other perfectly.

Exam Tip: A rectangle has only one horizontal line of symmetry (or vertical, depending on orientation) - not two like a square.

Question 5. (ii) Isosceles triangle
Answer: An isosceles triangle has one line of symmetry running from the top vertex down through the midpoint of the base, dividing it into two identical halves.
In simple words: A line from the point at the top to the middle of the bottom edge splits it into two matching parts.

Exam Tip: The line of symmetry in an isosceles triangle always goes through the vertex at the top and the midpoint of the opposite side.

Question 5. (iii) Rhombus
Answer: A rhombus has two lines of symmetry - both run along its diagonals, from one vertex to the opposite vertex, dividing it into two identical halves.
In simple words: Draw lines from the pointed corners to the opposite pointed corners - each line splits the shape into matching left and right halves.

Exam Tip: Both lines of symmetry in a rhombus are its two diagonals - they always cross at right angles at the centre.

Question 5. (iv) Pentagon
Answer: A regular pentagon has five lines of symmetry - each one runs from a vertex through the midpoint of the opposite side, dividing it into two identical halves.
In simple words: Draw five lines, each from a corner through the middle of the opposite edge - each line splits the pentagon into two matching parts.

Exam Tip: A regular pentagon always has exactly five lines of symmetry, one for each vertex.

Question 6. We get the symmetry if we shade according to the after diagonal with the same figure. Also we get the same figure if we shade by taking the lines joining mid points of opposite sides. Yes the figure is symmetrical about both diagonals.
Answer: When you shade a 4 x 4 grid following one diagonal pattern and then flip or rotate it, the result matches the original figure. Similarly, if you shade by following lines that join the midpoints of opposite sides, you create the same pattern. This works because the figure has symmetry about both its diagonals, meaning each diagonal acts as a line of reflection that creates identical patterns on both sides.
In simple words: The shading pattern looks the same whether you reflect it across the diagonal or across the lines connecting opposite midpoints.

Exam Tip: When checking symmetry in grid patterns, always verify that the shaded regions match exactly on both sides of the line of symmetry.

Question 7. (i) Angle bisector or Median
Answer: An angle bisector divides an angle into two equal parts. A median connects a vertex of a triangle to the midpoint of the opposite side. Both of these lines often serve as lines of symmetry in geometric figures.
In simple words: An angle bisector cuts an angle in half, and a median goes from a corner to the middle of the opposite edge.

Exam Tip: In an isosceles or equilateral triangle, the angle bisector, median, and line of symmetry from a vertex are all the same line.

Question 7. (ii) Diagonal
Answer: A diagonal connects two non-adjacent vertices of a polygon. In shapes like rhombuses, squares, and kites, diagonals often serve as lines of symmetry.
In simple words: A diagonal is a line drawn from one corner of a shape to another corner that is not next to it.

Exam Tip: Not all diagonals are lines of symmetry - check whether the shape folds exactly onto itself along the diagonal.

Question 7. (iii) Diameter of Circle
Answer: A diameter is a straight line that passes through the centre of a circle, connecting two points on the circle's edge. Every diameter of a circle acts as a line of symmetry, dividing the circle into two identical semicircles.
In simple words: Any line drawn through the middle of a circle from edge to edge splits it into two matching halves.

Exam Tip: A circle has infinite lines of symmetry - every single diameter is a line of symmetry, which is unique to circles among common shapes.

Exercise 14(2)

Question 1. (i) Rotational symmetry of order 2
Answer: A figure has rotational symmetry of order 2 when it appears identical after being turned 180 degrees around its centre point.
In simple words: The shape looks the same if you spin it halfway around.

Exam Tip: Order 2 means the figure matches itself at exactly 180 degrees - this is the most common rotational symmetry in rectangles and parallelograms.

Question 1. (ii) Rotational symmetry of order 2
Answer: A figure has rotational symmetry of order 2 when it appears identical after being turned 180 degrees around its centre point.
In simple words: The shape looks the same if you spin it halfway around.

Exam Tip: Order 2 means the figure matches itself at exactly 180 degrees - this is the most common rotational symmetry in rectangles and parallelograms.

Question 1. (iii) No rotational symmetry
Answer: This figure does not have rotational symmetry. It does not look the same when rotated by any angle less than 360 degrees around its centre.
In simple words: No matter which direction you spin this shape, it never looks exactly like the original until you turn it all the way around.

Exam Tip: If a figure has no rotational symmetry, it has only an order of 1 (which means it matches only when rotated the full 360 degrees).

Question 1. (iv) Rotational symmetry of order 2
Answer: A figure has rotational symmetry of order 2 when it appears identical after being turned 180 degrees around its centre point.
In simple words: The shape looks the same if you spin it halfway around.

Exam Tip: Order 2 means the figure matches itself at exactly 180 degrees - this is the most common rotational symmetry in rectangles and parallelograms.

Question 1. (v) No rotational symmetry
Answer: This figure does not have rotational symmetry. It does not look the same when rotated by any angle less than 360 degrees around its centre.
In simple words: No matter which direction you spin this shape, it never looks exactly like the original until you turn it all the way around.

Exam Tip: If a figure has no rotational symmetry, it has only an order of 1 (which means it matches only when rotated the full 360 degrees).

Question 1. (vi) Rotational symmetry of order 3
Answer: A figure has rotational symmetry of order 3 when it appears identical after being turned 120 degrees around its centre point. This means it matches itself three times as you rotate it through 360 degrees.
In simple words: The shape looks the same when you spin it by 120 degrees, 240 degrees, and 360 degrees.

Exam Tip: Order 3 is found in equilateral triangles and three-pointed star shapes - the shape repeats itself exactly three times.

Question 1. (vii) Rotational symmetry of order 3
Answer: A figure has rotational symmetry of order 3 when it appears identical after being turned 120 degrees around its centre point. This means it matches itself three times as you rotate it through 360 degrees.
In simple words: The shape looks the same when you spin it by 120 degrees, 240 degrees, and 360 degrees.

Exam Tip: Order 3 is found in equilateral triangles and three-pointed star shapes - the shape repeats itself exactly three times.

Question 1. (viii) No rotational symmetry
Answer: This figure does not have rotational symmetry. It does not look the same when rotated by any angle less than 360 degrees around its centre.
In simple words: No matter which direction you spin this shape, it never looks exactly like the original until you turn it all the way around.

Exam Tip: If a figure has no rotational symmetry, it has only an order of 1 (which means it matches only when rotated the full 360 degrees).

Question 1. (ix) Rotational symmetry of order 2
Answer: A figure has rotational symmetry of order 2 when it appears identical after being turned 180 degrees around its centre point.
In simple words: The shape looks the same if you spin it halfway around.

Exam Tip: Order 2 means the figure matches itself at exactly 180 degrees - this is the most common rotational symmetry in rectangles and parallelograms.

Question 1. (x) Rotational symmetry of order 3
Answer: A figure has rotational symmetry of order 3 when it appears identical after being turned 120 degrees around its centre point. This means it matches itself three times as you rotate it through 360 degrees.
In simple words: The shape looks the same when you spin it by 120 degrees, 240 degrees, and 360 degrees.

Exam Tip: Order 3 is found in equilateral triangles and three-pointed star shapes - the shape repeats itself exactly three times.

Question 1. (xi) Rotational symmetry of order 6
Answer: A figure has rotational symmetry of order 6 when it appears identical after being turned 60 degrees around its centre point. This means it matches itself six times as you rotate it through 360 degrees.
In simple words: The shape looks the same each time you turn it by 60 degrees - a perfect pattern that repeats six times going around.

Exam Tip: Order 6 is seen in regular hexagons and certain flower or star patterns - look for shapes that repeat naturally six times.

Question 1. (xii) Rotational symmetry of order 3
Answer: A figure has rotational symmetry of order 3 when it appears identical after being turned 120 degrees around its centre point. This means it matches itself three times as you rotate it through 360 degrees.
In simple words: The shape looks the same when you spin it by 120 degrees, 240 degrees, and 360 degrees.

Exam Tip: Order 3 is found in equilateral triangles and three-pointed star shapes - the shape repeats itself exactly three times.

Question 2. (1) and (ii) have rotational symmetry of order greater than 1.
Answer: This statement is correct. Both figures from question 1 parts (i) and (ii) display rotational symmetry at an order above 1, meaning they match themselves when rotated at specific angles before completing a full 360-degree turn.
In simple words: Both of these shapes look the same when rotated by less than a full spin around.

Exam Tip: When you see "order greater than 1," it simply means the figure has some rotational symmetry - look for any angle that makes it match itself before the full rotation.

Question 3. Rhombus and equilateral triangle have both line of symmetry and rotational symmetry.
Answer: A rhombus has two lines of symmetry (its diagonals) and rotational symmetry of order 2 (it looks the same after a 180-degree turn). An equilateral triangle has three lines of symmetry and rotational symmetry of order 3 (it looks the same after a 120-degree turn). Both shapes combine line symmetry with rotational symmetry.
In simple words: Both shapes can be folded in half along certain lines, and both look the same when spun around.

Exam Tip: Remember that not all shapes with line symmetry have rotational symmetry - check both properties separately for each figure.

Question 4. Rectangle, Rhombus and Square have both line of symmetry and rotational symmetry of order more than 1.
Answer: A rectangle has two lines of symmetry and rotational symmetry of order 2. A rhombus has two lines of symmetry and rotational symmetry of order 2. A square has four lines of symmetry and rotational symmetry of order 4. All three shapes display both types of symmetry, and all have rotational order greater than 1.
In simple words: Each of these shapes can be folded along lines to match itself, and each spins to match itself at least twice before going all the way around.

Exam Tip: The square is the most symmetric of these three - it has the most lines of symmetry and the highest rotational order.

Question 5. (i) Equilateral triangle
Answer: An equilateral triangle has three lines of symmetry. Each line runs from a vertex down through the midpoint of the opposite side. The triangle also has rotational symmetry of order 3 (it looks identical when rotated 120 degrees).
In simple words: All three sides are equal in length, and the shape has three fold lines and can be spun three times before looking the same.

Exam Tip: An equilateral triangle is one of the most symmetric triangles - it has three equal sides and three equal angles of 60 degrees each.

Question 5. (ii) Isosceles triangle
Answer: An isosceles triangle has one line of symmetry running from the top vertex down through the midpoint of the base, dividing it into two identical halves.
In simple words: A line from the point at the top to the middle of the bottom edge splits it into two matching parts.

Exam Tip: The line of symmetry in an isosceles triangle always goes through the vertex at the top and the midpoint of the opposite side.

Question 5. (iii) Scalene triangle
Answer: A scalene triangle has no lines of symmetry. All three sides have different lengths, and no line can divide it into two identical matching halves.
In simple words: All three sides are different lengths, so no fold line can make the two halves match.

Exam Tip: In a scalene triangle, every side and every angle is different from the others, which is why it has zero symmetry.

Question 5. (iv) Parallelogram
Answer: A parallelogram has no lines of symmetry. Although opposite sides are equal and parallel, no line through the figure can divide it into two identical matching halves. It does have rotational symmetry of order 2 (it looks the same when rotated 180 degrees).
In simple words: You cannot fold a parallelogram along any line to make the two halves match exactly, but it does look the same when spun halfway around.

Exam Tip: A parallelogram has rotational symmetry but no line symmetry - this is an important distinction to remember.

Question 5. (v) Isosceles trapezium
Answer: An isosceles trapezium has one line of symmetry running vertically through its centre, dividing it into two identical mirror halves. The two non-parallel sides are equal in length.
In simple words: A vertical line through the middle splits this trapezium into two matching left and right halves.

Exam Tip: The isosceles trapezium is the only trapezium with a line of symmetry - regular trapeziums have no symmetry at all.

Question 6. (i) Diagonal
Answer: A diagonal connects two non-adjacent vertices of a polygon. In grid diagrams shown in (i) through (vi), some diagonals act as lines of reflection symmetry for certain patterns and shapes.
In simple words: A diagonal is a line from one corner to another corner that is not next to it.

Exam Tip: Always check if a diagonal actually creates two identical matching halves before calling it a line of symmetry.

Question 6. (ii) Vertical or Horizontal line through the centre
Answer: A vertical or horizontal line passing through the centre of a figure acts as a line of reflection symmetry when both sides of the line are mirror images of each other.
In simple words: A line running straight up and down or straight left and right through the middle can be a fold line if the shape matches on both sides.

Exam Tip: Check your grid pattern carefully to see if the left and right halves (or top and bottom halves) are exact mirrors of each other.

Question 7. \( 120°, 180°, 240°, 300°, 360° \)
Answer:
(i) \( 144°, 216°, 288° \) and \( 360° \)
(ii) \( 90°, 135°, 180°, 225°, 270°, 315° \) and \( 360° \)
(iii) Angle of rotation of \( 50° \) is not possible

In simple words: A shape can be rotated by certain special angles and still look the same. The angle must divide evenly into 360 degrees.

Exam Tip: To find angles of rotation, divide 360 by the order of rotational symmetry - if order is 3, then 360 ÷ 3 = 120 degrees.

Question 8. (i) Yes, i.e 180 and 360°
Answer: Yes, this figure has rotational symmetry. It looks the same after rotating 180 degrees and again after a full 360-degree rotation.
In simple words: The shape matches itself when you turn it halfway around and again when you turn it all the way.

Exam Tip: If a figure looks identical at 180 degrees, it has at least order 2 rotational symmetry.

Question 8. (ii) Yes i.e 120, 240 and 360°
Answer: Yes, this figure has rotational symmetry. It looks the same after rotating 120 degrees, 240 degrees, and 360 degrees, indicating rotational symmetry of order 3.
In simple words: The shape matches itself at three different turns before going all the way around.

Exam Tip: Figures with 120-degree rotational angles have order 3 symmetry - think of equilateral triangles and three-pointed designs.

Question 8. (iii) Yes i.e 90, 180, 270 and 360°
Answer: Yes, this figure has rotational symmetry. It looks the same after rotating 90 degrees, 180 degrees, 270 degrees, and 360 degrees, indicating rotational symmetry of order 4.
In simple words: The shape matches itself at four different turns - every quarter turn, half turn, and three-quarter turn.

Exam Tip: Figures with 90-degree rotational angles have order 4 symmetry - squares and four-pointed stars are good examples.

Question 8. (iv) Yes i.e 30, 60, 90, 120, 150, 180, 210, 240, 270, 300° & 360°
Answer: Yes, this figure has rotational symmetry. It matches itself when rotated by every 30-degree increment up to 360 degrees, indicating rotational symmetry of order 12.
In simple words: This shape has very high symmetry - it repeats its pattern twelve times as you spin it around.

Exam Tip: Figures with 30-degree rotational angles have order 12 symmetry - twelve-petalled flowers or complex star patterns show this level of symmetry.

Question 8. (v) Yes i.e 15,30, 45, 60, 25, 90, 105, 120, 135, 150, 165, 180, 195, 210, 225, 240, 255, 270, 285, 300, 315, 330, 345, 360°
Answer: Yes, this figure has rotational symmetry. It matches itself when rotated by every 15-degree increment up to 360 degrees, indicating rotational symmetry of order 24.
In simple words: This shape has extremely high symmetry - it repeats its pattern 24 times as you turn it around completely.

Exam Tip: Figures with 15-degree rotational angles have order 24 symmetry - this is exceptional symmetry found only in highly regular patterns like complex mandalas or mathematical designs.

Question 8. (vi) No
Answer: No, this figure does not have rotational symmetry. It does not look the same when rotated by any angle less than 360 degrees.
In simple words: You have to turn this shape all the way around to see it match itself again.

Exam Tip: When a figure has no rotational symmetry except at 360 degrees, it is said to have order 1 rotational symmetry.