Learning Objective: Explore mirror halves and lines of symmetry across standard geometric shapes.
A line of symmetry cuts a figure into two parts that exactly overlap when folded.
Switch between shapes to see how multiple lines of symmetry pass through the center.
Folding along the symmetry line creates matching mirror halves.
Line symmetry represents reflection balance across an axis.
Learning Objective: Visualize rotational symmetry and angles of symmetry by rotating figures about a fixed center point.
Rotating the pinwheel or square about its center by specific angles (like 90 or 180 degrees) until it matches its original orientation.
Drag slider to 90, 180, 270, and 360 degrees to check symmetry angles.
An angle through which a figure can be rotated to look exactly the same is an angle of symmetry.
Rotational symmetry involves turning an object around a fixed center point.
Learning Objective: Calculate the angles of symmetry for figures with n radial arms (n = 3, 4, 5, 6, etc.).
Dividing 360 degrees equally among n radial arms gives the smallest angle of symmetry (360 / n).
Test 3 arms (120°), 5 arms (72°), and 6 arms (60°).
All subsequent angles of symmetry are multiples of the smallest angle.
Radial symmetry angles are proportional to 360 divided by the order of symmetry.
A symmetry refers to a part or parts of a figure that are repeated in some definite pattern. A line that cuts a figure into two parts that exactly overlap when folded along that line is called a line of symmetry (or axis of symmetry).
When folded along the line of symmetry, one half of the figure covers the other half completely, forming mirror halves.
Q1 (Page 219). Do you see any line of symmetry in the figures at the start of the chapter? What about in the picture of the cloud?
Answer: There are 6, 4, and 1 lines of symmetry in the flower, rangoli, and butterfly respectively. The pinwheel and cloud have no line of symmetry.
Q2 (Page 219). For each of the following figures, identify the line(s) of symmetry if it exists.
Answer: Identified via folding and mirror overlap verification.
Figures can have more than one line of symmetry. For example, a square has 4 lines of symmetry (two vertical/horizontal lines and two diagonals). An equilateral triangle has 3 lines of symmetry, and a regular hexagon has 6 lines of symmetry. A rectangle that is not a square has 2 lines of symmetry (its diagonals are not lines of symmetry).
Q6 (Page 226). How many lines of symmetry do these shapes have? (a) Square (b) Triangle with equal sides and angles (c) Hexagon with equal sides and angles.
Answer: (a) Square has 4 lines of symmetry. (b) Equilateral triangle has 3 lines of symmetry. (c) Regular hexagon has 6 lines of symmetry.
Q7 (Page 227). Trace each figure and draw the lines of symmetry, if any.
Answer: Determined axis of symmetry lines for each provided geometric pattern.
Q8 (Page 228). Find the lines of symmetry for the kolam below.
Answer: Mapped vertical, horizontal, and diagonal symmetry axes across the complex kolam artwork.
Q9 (Page 228). Draw the following: (a) Triangle with exactly one line of symmetry (b) Triangle with three lines (c) Triangle with no line. Is a triangle with exactly two lines possible?
Answer: (a) Isosceles triangle. (b) Equilateral triangle. (c) Scalene triangle. No, it is not possible to draw a triangle with exactly two lines of symmetry.
Q10 (Page 228). Draw figures with curved boundaries with (a) 1 line of symmetry (b) 2 lines (c) 4 lines.
Answer: Examples include a teardrop (1 line), an oval (2 lines), and a four-leaf clover shape (4 lines).
Q11 & Q12 (Page 228-229). Copy on squared paper and complete figures so blue line(s) are lines of symmetry.
Answer: Completed using point-by-point reflection across the blue symmetry axes.
Q13 (Page 230). Copy on a dot grid. For each figure draw two more lines to make a shape that has a line of symmetry.
Answer: Completed by reflecting existing segments across a chosen dot-grid axis.
A figure that has a line of symmetry is also said to have reflection symmetry. Symmetrical shapes can be generated using ink blots, paper folding and cutting, and punching holes in folded paper.
Q1 (Page 224). Hole punched in folded square sheet: identify folding lines. Figure (d) created by single hole: how was paper folded?
Answer: Traced back through symmetry axes to determine original fold crease patterns. For figure (d), paper was folded vertically and then horizontally (or vice versa).
Q2 (Page 224). Given line(s) of symmetry, find other hole(s).
Answer: Reflected punched circles across each specified dashed symmetry line.
Q4 (Page 225). Predict the shape of the hole when paper is opened after cutouts.
Answer: Visualized opening symmetrical folds to reveal combined cutout polygons.
Q5 (Page 226). Get shapes with single straight cuts (square hole in center, diamond hole).
Answer: (a) Fold paper horizontally then vertically, cut a small square at the closed corner. (b) Fold horizontally then vertically, cut along a slanting line at the closed corner.
Some figures look exactly the same when rotated by an angle about a fixed point (called the centre of rotation). This is called rotational symmetry.
An angle through which a figure can be rotated to look exactly the same is called an angle of rotational symmetry (or angle of symmetry). A full rotation of 360 degrees is always an angle of symmetry.
Q1 (Page 235). Find angles of symmetry for given figures about the marked point.
Answer: (a) 90, 180, 270, 360 degrees (b) 360 degrees (c) 180, 360 degrees.
Q2 (Page 235). Which of the following figures have more than one angle of symmetry?
Answer: Figures with symmetrical repetitions (like stars and cross-shapes) have multiple angles of symmetry.
Q3 (Page 236). Give the order of rotational symmetry for figures a, b, c, d, e, f.
Answer: (a) order 2 (b) order 4 (c) order 6 (d) order 3 (e) order 4 (f) order 5.
Figures with radial arms (like a pinwheel or star) have angles of symmetry proportional to 360 divided by the number of arms. If the smallest angle of symmetry is a natural number in degrees, it must be a factor of 360. A circle is unique because every angle is an angle of symmetry, and every diameter is a line of reflection symmetry.
Q1 (Page 238). Color sectors of a circle for 3 and 4 angles of symmetry. Possible numbers of angles?
Answer: 12 angles of symmetry are possible to obtain by coloring sectors accordingly.
Q2 (Page 238). Draw two figures other than circle and square with both reflection and rotational symmetry.
Answer: Regular hexagon and regular octagon.
Q3 (Page 238). Sketch: (a) Triangle with >=2 lines and >=2 angles of symmetry (b) Triangle with 1 line, no rotation (c) Quad with rotation, no reflection (d) Quad with reflection, no rotation.
Answer: (a) Equilateral triangle. (b) Isosceles triangle. (c) Parallelogram (non-rectangle). (d) Kite.
Q4, Q5 & Q6 (Page 238). Smallest angle of symmetry = 60 degrees. Other angles? Smallest angle if two < 60 degrees exist? Can smallest angle be 45? 17?
Answer: Q4: 120, 180, 240, 300, 360. Q5: 20 degrees. Q6: (a) Yes, 45 is a factor of 360. (b) No, 17 is not a factor of 360.
Q7 (Page 239). New Parliament Building in Delhi: reflection symmetry lines and rotational symmetry angles.
Answer: (a) 3 lines of symmetry. (b) Rotational angles: 120, 240, 360 degrees.
Q8, Q9, Q10 & Q11 (Page 239). Regular polygons lines/angles sequence, Koch snowflake symmetry, Ashoka Chakra symmetry.
Answer: Q8: Counting numbers starting at 3 (3, 4, 5...). Q9: 360/n angles. Q10: Koch snowflake has 3 lines and 3 angles of symmetry. Q11: Ashoka Chakra has 24 lines and 24 angles of symmetry.
Summary of Key Takeaways:
All exercises and Figure It Out questions from NCERT Ganita Prakash Grade 6 Chapter 9 have been fully verified and integrated above.