Symmetry
Complete, step-by-step solutions — every in-text question and every “Figure it Out” exercise, with clear diagrams and the lines & angles of symmetry drawn in.
In-text Questions
Discussion & “Math Talk” prompts worked through as you read the chapter.
9.1 Line of Symmetry Page 221–222
Is there any other way to fold the square so that the two halves overlap? How many lines of symmetry does the square shape have?
Fold a square along the vertical, the horizontal and each of the two diagonals — every one of these folds makes the two halves overlap exactly. Trying any other fold, the halves no longer match.
No other fold works · A square has 4 lines of symmetry
We saw that the diagonal of a square is a line of symmetry. Take a rectangle that is not a square. Is its diagonal a line of symmetry?
Folding a non-square rectangle along a diagonal, the two triangular halves do not cover each other — the long side falls over the short side. So the diagonal is not a fold line.
No — a rectangle’s diagonal is not a line of symmetry
Reflecting the labelled square Page 222
What if we reflect the square (corners A, B, C, D) along the diagonal from A to C? Where do A, B, C, D go? And along the horizontal line of symmetry?
- Reflection in diagonal A→C: A and C lie on the line, so they stay put. B and D swap — D moves to where B was, and B moves to where D was.
- Reflection in the horizontal line: the top edge swaps with the bottom edge. A↔D and B↔C — so D takes A’s place and C takes B’s place.
Diagonal A→C: A, C fixed; B, D interchange · Horizontal line: A↔D, B↔C
9.2 Rotational Symmetry Page 235–236
Can you draw a radial-arm figure with (a) exactly 5 angles of symmetry, (b) 6 angles of symmetry? Find the angles in each case.
With n equal radial arms the smallest turn that maps the figure onto itself is \(\dfrac{360^\circ}{n}\), and every multiple of it up to \(360^\circ\) is an angle of symmetry.
- 5 arms \(\left(\tfrac{360^\circ}{5}=72^\circ\right)\): 72°, 144°, 216°, 288°, 360°
- 6 arms \(\left(\tfrac{360^\circ}{6}=60^\circ\right)\): 60°, 120°, 180°, 240°, 300°, 360°
Consider a radial-arm figure with exactly 7 angles of symmetry. What is its smallest angle of symmetry? Is it a whole number of degrees? If not, write it as a mixed fraction.
The smallest angle is \(\dfrac{360^\circ}{7}\). Dividing, \(360=7\times 51+3\), so \(\dfrac{360}{7}=51\tfrac{3}{7}\).
Smallest angle \(=\dfrac{360^\circ}{7}=51\tfrac{3}{7}^{\circ}\) — not a whole number
In each case the angles of symmetry were multiples of the smallest one. Will this always happen?
Yes. If a turn of angle \(a\) leaves the figure unchanged, then doing it twice (turn \(2a\)), three times (\(3a\)), … also leaves it unchanged. So every angle of symmetry is a multiple of the smallest one.
Yes — always multiples of the smallest angle
- Every figure has 360° as an angle of symmetry. → True (a full turn brings any figure back to itself).
- If the smallest angle of symmetry is a whole number of degrees, it is a factor of 360. → True (the multiples must land exactly on 360°).
Both statements are True
Figure it Out — Exercises
Every exercise question from Sections 9.1 and 9.2, solved with diagrams.
9.1 Line of Symmetry Page 219
Do you see any line of symmetry in the figures at the start of the chapter? What about the cloud?
Flower — 6 · Rangoli — 4 · Butterfly — 1 · Pinwheel & Cloud — none
For each of the following figures, identify the line(s) of symmetry if any.
The arrow, the kite and the L-shaped figure each have exactly one line of symmetry (drawn dashed). The slanted quadrilateral and the scalene triangle have no line of symmetry.
Arrow, kite, L-shape → 1 line each · other two → none
9.1 Punching & Paper-Cutting Page 223–229
A hole was punched in a folded square sheet, then unfolded. Identify the fold line. Figure (d) has a single hole — how was it folded?
The fold line is the mirror line halfway between each pair of holes. In (d) the four holes need two folds — vertical then horizontal (or the reverse) — a single punch then opens into four.
a — vertical · b — diagonal · c — horizontal · d — folded vertically then horizontally
Given the line(s) of symmetry, find the other hole(s).
Each new hole is the reflection of the given hole in the line — the same perpendicular distance on the opposite side. A figure with two lines produces holes on both sides.
Reflect every given hole across each line of symmetry
Predict the shape of the hole when the paper is opened after each cut.
The cut is mirrored across each fold. With a single vertical fold (a, b) the notch you cut is doubled left–right, giving a symmetric hole. In the two-fold cases (c, d) the notch is doubled in both directions: (c) opens into a four-fold “brick/plus” pattern of square notches, and (d) opens into an I / H-shaped hole.
a — bow-tie · b — hexagonal ribbon · c — 4-fold square-notch pattern · d — I-shaped hole
Get each shape with some folds and a single straight cut. (a) central square hole; (b) central square (tilted like a diamond) hole.
(a) Fold the sheet in half horizontally, then in half again vertically. At the corner where all folded layers meet (the centre of the sheet), cut a small square with all sides closed. Opening it gives a square hole in the centre.
(b) Fold the same way. This time make a single slanting cut across the closed corner. Opening it gives a tilted square (diamond) hole in the centre.
Check: in both, the four-sided hole has equal sides and right angles, so it is a genuine square.
Fold in half twice (⟂ folds); (a) cut a small square at the centre corner · (b) cut a slant at the centre corner
How many lines of symmetry do these shapes have?
Square — 4 · 8-point star — 8 · Equilateral triangle — 3 · Regular hexagon — 6
Trace each figure and draw the lines of symmetry, if any.
Draw every mirror line through the centre — counts shown above each figure
Find the lines of symmetry for the kolam below.
The star pattern repeats every 60°, so mirror lines run through each of the 6 points and through each of the 6 gaps between them.
The kolam has 6 lines of symmetry
Draw a triangle with (a) exactly one, (b) exactly three, (c) no line of symmetry. Can a triangle have exactly two lines of symmetry?
A triangle with two lines of symmetry would need two pairs of equal sides — but that forces all three sides equal, giving three lines. So two is impossible.
(a) isosceles · (b) equilateral · (c) scalene · Exactly two lines — not possible
Draw figures (each with at least one curved boundary) having exactly (a) one, (b) two, (c) four lines of symmetry.
Curved figures with exactly 1, 2 and 4 lines of symmetry
Copy on squared paper and complete each figure so that the blue line is a line of symmetry. (Problem (a) is done for you.)
For each figure, reflect the given red outline across the blue line — every corner moves to the mirror-image square the same distance on the other side. Joining the reflected corners completes the symmetric figure. For the slanting blue lines in (c) and (f), turning the page so the line is vertical makes the reflection easier to plot.
Reflect the red shape across the blue line to complete each figure
Complete each drawing so that the resulting figure has both blue lines as lines of symmetry.
First reflect the given shape across the first blue line. Then reflect everything you now have (original + new part) across the second blue line. With two perpendicular lines this produces four matching copies arranged around the crossing point.
Reflect across both blue lines — you get four matching copies
On a dot grid, draw two more lines on each figure to make a shape that has a line of symmetry.
Pick a mirror line, then add two segments that are the reflections of the segments already drawn. The open figure closes up into a symmetric shape (many correct answers are possible).
Add the two mirror-image segments to close a symmetric shape
9.2 Angles of Symmetry Page 235–236
Find the angles of symmetry for the given figures about the marked point.
- (a) four equal arms → 90°, 180°, 270°, 360°
- (b) arms unequal / one-sided → only 360° (a full turn)
- (c) two opposite equal arms → 180°, 360°
Which of the figures have more than one angle of symmetry?
A figure has more than one angle of symmetry when it maps onto itself for some turn smaller than a full 360°.
The circle-with-cross, 3-sector circle, pinwheel, crossed-lines and 5-point star all qualify
Give the order of rotational symmetry for each figure.
The order is how many times the figure looks the same in one full turn — that is, \(360^\circ \div (\text{smallest angle})\).
| Figure | (a) S-shape | (b) crossed lines | (c) 6-star | (d) running arms | (e) plus | (f) pentagon |
|---|---|---|---|---|---|---|
| Order | 2 | 4 | 6 | 3 | 4 | 5 |
In each case the angles were multiples of the smallest. Does this always happen?
Yes — repeating a symmetry turn gives the next angle, so the second is twice the first, the third is three times, and so on. All angles of symmetry are multiples of the smallest one.
Yes, always multiples of the smallest angle
9.2 Circles, Polygons & Structures Page 238–239
Colour the 12 sectors so the figure has (i) 3, (ii) 4 angles of symmetry. (iii) What numbers of angles of symmetry are possible?
A colouring has \(k\) angles of symmetry when the pattern repeats every \(\tfrac{12}{k}\) sectors — so \(k\) must divide 12.
(i) 3 · (ii) 4 · (iii) possible values are the divisors of 12: 1, 2, 3, 4, 6, 12
Draw two figures (other than a circle and a square) that have both reflection and rotational symmetry.
e.g. a plus-shape and a 4-petal flower — each has 4 lines and rotational order 4
Draw, where possible: (a) a triangle with ≥2 lines and ≥2 angles of symmetry; (b) a triangle with one line but no rotational symmetry; (c) a quadrilateral with rotational but no reflection symmetry; (d) a quadrilateral with reflection but no rotational symmetry.
In a figure, 60° is the smallest angle of symmetry. What are the other angles?
All angles are multiples of 60° up to 360°.
120°, 180°, 240°, 300°, 360°
60° is an angle of symmetry, and the figure has two angles of symmetry smaller than 60°. What is the smallest?
The smallest angle \(s\) must divide 60°, and exactly two of its multiples fall below 60°. Those two are \(s\) and \(2s\), with \(3s=60^\circ\). So \(s = 20^\circ\) (giving 20°, 40°, 60°, …).
Smallest angle of symmetry = 20°
Can a figure have rotational symmetry whose smallest angle is (a) 45°, (b) 17°?
The smallest angle must be a factor of 360°. \(360\div45=8\) (whole), but \(360\div17\) is not a whole number.
- (a) 45° — Yes, since 360° is a multiple of 45°.
- (b) 17° — No, since 360° is not a multiple of 17°.
(a) Yes · (b) No
The new Parliament Building in Delhi — (a) does its outer boundary have reflection symmetry? (b) rotational symmetry about its centre?
The outline is a triangle-like (three-fold) shape.
- (a) Yes — it has 3 lines of symmetry.
- (b) Yes — angles of rotational symmetry are 120°, 240°, 360°.
3 lines of symmetry · rotation angles 120°, 240°, 360°
How many lines / angles of symmetry do the regular polygons (Chapter 1, Table 3) have? What number sequence do you get?
A regular polygon with \(n\) sides has exactly \(n\) lines of symmetry and \(n\) angles of symmetry.
| Polygon | Triangle | Quadrilateral | Pentagon | Hexagon | Heptagon | Octagon | Nonagon | Decagon |
|---|---|---|---|---|---|---|---|---|
| Lines = Angles | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
The counting-number sequence 3, 4, 5, 6, 7, 8, 9, 10, …
How many lines and angles of symmetry do the shapes in the Koch Snowflake sequence (Chapter 1, Table 3) have?
The first shape is a triangle (3 of each). From the next stage on, the snowflake keeps its six-fold shape, so it has 6 each at every later stage.
Lines of symmetry: 3, 6, 6, 6, 6 · Angles of symmetry: 3, 6, 6, 6, 6
How many lines of symmetry and angles of symmetry does the Ashoka Chakra have?
The Chakra has 24 equally-spaced spokes, so a mirror line runs through each spoke and it maps onto itself after every \(\tfrac{360^\circ}{24}=15^\circ\) turn.
24 lines of symmetry · 24 angles of symmetry