Chapter 9 exercise answers: Symmetry

Open NCERT p. 219Matches NCERT’s answer

1. The flower's petals repeat evenly all the way round, so a fold through the middle of any petal lands the two halves on each other -- yes.
2. The butterfly's two wings are mirror images of each other about its body line -- yes.
3. The rangoli's four coloured quarters repeat around its centre and mirror each other -- yes.
4. The pinwheel's blades all lean the same rotational way, so no fold line ever lands one blade where another already is -- no.
5. The cloud has no repeated part at all, so no fold line works -- no.

AnswerFlower: yes. Butterfly: yes. Rangoli: yes. Pinwheel: no. Cloud: no.

Watch this explained “The test run back over the openers — and over the clouds”, 5:53 into A line of symmetry is a fold that makes the halves coincide · हिंदी में देखें

Open NCERT p. 219Checked by computer

1. The 1st figure (a dart with a notch) folds along the line straight through its point and the notch -- 1 line.
2. The 2nd figure (a kite) folds only along its long diagonal -- 1 line.
3. The 3rd figure has an upright side and a slanted top with no matching sides at all -- 0 lines.
4. The 4th figure (an L with two equal arms) folds along the diagonal joining its two outer corners, on the slant -- 1 line.
5. The 5th figure (a long thin triangle) has three different side lengths -- 0 lines.

Answer1, 1, 0, 1, 0 lines of symmetry respectively.

Watch this explained “Five plain outlines, and no opinion required for any of them”, 7:29 into A line of symmetry is a fold that makes the halves coincide · हिंदी में देखें

Open NCERT p. 224Matches NCERT’s answer

1. (a) The two holes are at the same height, one near the left edge and one near the right edge. The fold has to carry one hole onto the other, so it is the upright line down the middle of the sheet.
2. (b) The two holes are close together near the top-right corner, one a little above and to the left of the other. The line that carries one onto the other runs from the top-right corner to the bottom-left corner, so the fold is that diagonal.
3. (c) The two holes are the same distance from the right edge, one above the middle of the sheet and one the same distance below it. The fold is the flat line across the middle.
4. (d) One punch through a sheet folded once makes only 2 holes. 4 holes from a single punch need 4 layers, so the sheet was folded twice.
5. The 4 holes are mirror images across the upright middle line and across the flat middle line, so the sheet was folded in half upright and then in half across (either order), and punched once near a corner of the small folded square.

Answer(a) Along the upright line down the middle. (b) Along the diagonal from the top-right corner to the bottom-left corner. (c) Along the flat line across the middle. (d) Folded twice, in half upright and then in half across (into quarters), and punched once.

Watch this explained “Reading the fold backwards from where the holes came out”, 4:00 into Fold, blot, cut, punch: making a figure that is symmetric by construction · हिंदी में देखें

Open NCERT p. 224Matches NCERT’s answer

1. In each figure one hole is marked and it is not on the dashed line of symmetry. Its partner is straight across the line, at right angles to it, and the same distance from the line on the other side. A hole off the line has exactly one partner, so each figure needs exactly 1 more hole.
2. (a) Square, with the diagonal from the top-left corner to the bottom-right corner: the hole is near the left edge, a little below the top-left corner. Its partner is near the top edge, a little to the right of that corner (the hole's distances from the left edge and from the top edge swap).
3. (b) Square, with the flat middle line: the hole is near the right edge, below the line. Its partner is straight above it, as far above the line as the hole is below it.
4. (c) Triangle, with the upright line through its top corner: the hole is just left of the line. Its partner is at the same height, the same distance right of the line.
5. (d) Circle, with a diameter rising to the right: the hole is just above the diameter, on the right side of the circle. Its partner is straight across, just below the diameter, still on the right side.
6. (e) Circle, with a diameter falling to the right: the hole is near the top, just above the diameter. Its partner is straight across, just below the diameter, towards the upper left of the circle.

AnswerExactly 1 more hole in each of (a) to (e), at the mirror position across the line: (a) near the top edge, just right of the top-left corner; (b) straight above the marked hole; (c) at the same height, just right of the line; (d) just below the diameter on the right; (e) just below the diameter, to the upper left.

Watch this explained “Marking the partner hole without folding anything”, 5:52 into Fold, blot, cut, punch: making a figure that is symmetric by construction · हिंदी में देखें

Open NCERT p. 224One way to think about it

1. A vertical fold means the paper is folded along an upright (top-to-bottom) line, so the left half swings onto the right half (or the right onto the left).
2. A horizontal fold means the paper is folded along a side-to-side line, so the top half swings down onto the bottom half.

In shortThese are just names for the two simplest folds: a vertical fold creases the paper upright (left-right halves meet), and a horizontal fold creases it across (top-bottom halves meet). Later questions build on these two folds, one after another.

Watch the lesson Fold, blot, cut, punch: making a figure that is symmetric by construction · हिंदी में देखें

Open NCERT p. 225Checked by computer

1. A cut through folded paper goes through every layer, so when the paper is opened the cut appears once for each layer, mirrored across each fold. A cut that touches a fold opens into one piece joined across that fold. A cut that touches only open edges opens into separate notches.
2. The book draws a vertical fold with the open edges on the right, so in (a), (b) and (d) the fold is the left edge of the folded sheet. In (c) the sheet is folded upright (fold on the left) and then folded down from the top (fold along the top), and both cuts are made in that small packet.
3. (a) Both ends of the zigzag cut lie on the fold. Opened, it is one hole in the middle of the sheet, across the fold and touching no edge: the cut shape joined to its mirror image, narrowest at the fold and wider towards its two zigzag ends.
4. (b) The flag-shaped cut starts and ends on the fold. Opened, it is one hole in the middle of the sheet, touching no edge: a band with a V-shaped notch at each end.
5. (c) The square cut on the left edge touches the upright fold but not the top fold. It opens into a rectangle twice as wide, and the top fold makes a second copy: two separate rectangular holes on the upright middle line, one above and one below the flat middle line.
6. (c) The square cut on the bottom edge touches only open edges, so it opens into 4 small notches: 2 in the top edge and 2 in the bottom edge of the sheet.
7. (d) The rectangle cut from the fold edge opens into one rectangular window in the centre. The rectangle cut from the open edge at the same height opens into two notches, one in each side edge. This is the book's own finished picture of (d).

Answer(a) One hole in the middle, across the fold, with a zigzag at each end. (b) One hole in the middle, a band with a V-shaped notch at each end. (c) Two rectangular holes on the upright middle line (one above and one below the centre) and 4 notches, 2 in the top edge and 2 in the bottom edge. (d) A tall rectangle with a notch in each side edge and a rectangular window in the centre.

Watch this explained “Predicting the opened figure before opening it”, 6:47 into Fold, blot, cut, punch: making a figure that is symmetric by construction · हिंदी में देखें

Open NCERT p. 226One way to think about it

a

1. Fold the square sheet in half along one diagonal, then in half along the other diagonal. The centre of the sheet is now a corner of the small folded triangle, where the two folds meet at a right angle.
2. Make one straight cut across that corner, at 45° to both folds. It cuts off a small right-angled triangle with two equal sides.
3. Open the sheet. The cut appears 4 times, once in each quarter, and the 4 copies form a hole in the centre whose sides are parallel to the sides of the sheet.
4. It is a square: its 4 sides are copies of the same cut, so they are equal, and each corner of the hole is made of two 45° angles, so it is 90°.

In shortFold along both diagonals, then make one straight cut across the folded centre corner at 45° to both folds. It opens into a square hole in the centre with sides parallel to the sheet's sides.

b

1. Fold the square sheet in half upright, then in half across. The centre of the sheet is now a corner of the small folded square, where the two folds meet at a right angle.
2. Make one straight cut across that corner, at 45° to both folds, cutting off a small right-angled triangle with two equal sides.
3. Open the sheet. The 4 copies of the cut form a square hole in the centre standing on one of its corners, with its corners pointing to the middles of the sheet's sides.
4. It is a square for the same reason: 4 equal copies of the cut for its sides, and corners of 45° + 45° = 90°.

In shortFold in half upright and then across, then make one straight cut across the folded centre corner at 45° to both folds. It opens into a square hole in the centre, turned onto its corner.

Watch this explained “A square hole two ways, with one straight cut”, 7:40 into Fold, blot, cut, punch: making a figure that is symmetric by construction · हिंदी में देखें

Open NCERT p. 226Matches NCERT’s answer

a

1. The first shape is a square standing on one corner -- still 4 equal sides and angles, so 4 lines.
2. The second shape is a regular eight-pointed star -- 8 equal points evenly spaced, so 8 lines.

Answer4 lines for the tilted square, 8 lines for the eight-pointed star.

b A triangle with equal sides and equal angles.

1. An equilateral triangle has 3 equal sides, one fold line through each vertex and the midpoint of the opposite side.

Answer3 lines of symmetry.

c A hexagon with equal sides and equal angles.

1. A regular hexagon has 6 equal sides, one fold line through each pair of opposite vertices or opposite edge-midpoints.

Answer6 lines of symmetry.

Watch this explained “Regular polygons: as many fold lines as sides”, 5:08 into Why a figure can have several lines of symmetry · हिंदी में देखें

Open NCERT p. 227Matches NCERT’s answer

1. Row 1, 1st figure (one diamond on top of two): only the upright line through the middle of the top diamond works. 1 line.
2. Row 1, 2nd figure (three diamonds in a row): the upright line through the middle diamond and the flat line through all three work. 2 lines.
3. Row 1, 3rd figure (a big diamond made of four small ones): its upright and flat diagonals work. 2 lines.
4. Row 1, 4th figure (two long strips crossing): the upright line and the flat line through the crossing work. 2 lines.
5. No slanted line works for any of these, because the small diamonds are taller than they are wide.
6. Row 2, top-left (a square with a smaller square and a cross inside, all centred): upright, flat and both diagonals work. 4 lines.
7. Row 2, top-right (an eight-sided figure 4 squares wide and 6 squares tall): the upright and flat lines work. It is taller than it is wide, so no diagonal works. 2 lines.
8. Row 2, bottom-left (a five-sided figure): the flat line through its left corner works. The top and bottom corners are 3 squares above and below that line, and the two right-hand corners are 1 square above and below it. No other line works. 1 line.
9. Row 2, bottom-right (a four-pointed star): upright, flat and both diagonals work. 4 lines.

AnswerRow 1: 1, 2, 2, 2 lines. Row 2: top-left 4, top-right 2, bottom-left 1, bottom-right 4.

Watch this explained “Three more figures, counted rather than admired”, 4:23 into Why a figure can have several lines of symmetry · हिंदी में देखें

Open NCERT p. 228Matches NCERT’s answer

1. The kolam is made of 7 identical six-pointed stars, one in the middle and six around it at the corners of a regular hexagon, with a small diamond in each gap between two neighbouring stars (12 diamonds).
2. A line through the middle star and two opposite outer stars is a line of symmetry: every star and diamond on one side lands on one on the other side. There are 3 such lines.
3. A line through the middle star and the gaps between opposite pairs of outer stars is also a line of symmetry. The upright line through the top and bottom diamonds is one. There are 3 such lines.
4. No other line works, because a fold has to send each outer star onto an outer star.
5. The dots are a drawing aid and are not placed evenly on the outermost tips in the printed picture. The count is for the design itself.

Answer6 lines of symmetry.

Watch this explained “Three more figures, counted rather than admired”, 4:23 into Why a figure can have several lines of symmetry · हिंदी में देखें

Open NCERT p. 228Matches NCERT’s answer

a A triangle with exactly one line of symmetry.

1. Take the triangle with corners (0, 0), (4, 0) and (2, 3).
2. Its two slanting sides are equal (each is √13 units long) and its base is 4 units, so it is isosceles but not equilateral.
3. Its only line of symmetry is the upright line x = 2, through the top corner and the middle of the base.

AnswerTriangle (0, 0), (4, 0), (2, 3): exactly 1 line of symmetry, the line x = 2.

b A triangle with exactly three lines of symmetry.

1. Take the triangle with corners (0, 0), (4, 0) and (2, 2√3).
2. All three sides are 4 units long, so it is equilateral.
3. It has 3 lines of symmetry, each through a corner and the middle of the opposite side.

AnswerTriangle (0, 0), (4, 0), (2, 2√3): exactly 3 lines of symmetry.

c A triangle with no line of symmetry.

1. Take the triangle with corners (0, 0), (6, 0) and (1, 4).
2. Its sides are 6, √17 and √41 units long, all different (a scalene triangle).
3. A fold line would have to make two sides equal, so it has no line of symmetry.

AnswerTriangle (0, 0), (6, 0), (1, 4): no line of symmetry.

d

1. A line of symmetry must send corners to corners. A triangle has 3 corners, an odd number, so the line keeps one corner where it is and swaps the other two.
2. Then the two sides meeting at the corner that stays are equal.
3. If a triangle had 2 lines of symmetry, it would have two different pairs of equal sides. The two pairs share a side, so all three sides would be equal.
4. But a triangle with all three sides equal has 3 lines of symmetry, not 2. So a triangle has 0, 1 or 3 lines of symmetry, never exactly 2.

AnswerNo, a triangle cannot have exactly 2 lines of symmetry.

Watch this explained “Triangles: three, one, or none”, 5:55 into Why a figure can have several lines of symmetry · हिंदी में देखें

Open NCERT p. 228Checked by computerAnswers can differ: one example

a A figure with exactly one line of symmetry.

1. A semicircle (half-disc) has one straight diameter edge and one curved arc edge.
2. Folding it along the diameter's perpendicular bisector lands the two halves on each other; no other fold works.

AnswerA semicircle: exactly one line of symmetry.

b A figure with exactly two lines of symmetry.

1. An ellipse (a non-circular oval) has two axes -- the long one and the short one.
2. Folding along either axis lands the halves on each other; a diagonal fold cannot, since it would have to swap the long radius with the short one.

AnswerAn ellipse: exactly two lines of symmetry.

c A figure with exactly four lines of symmetry.

1. A flower of four identical, evenly spaced curved petals repeats every quarter turn.
2. Folding through opposite petals works twice, and folding between opposite pairs of petals works twice more -- four lines in all.

AnswerA four-petalled flower: exactly four lines of symmetry.

Watch this explained “Curved boundaries count too”, 7:37 into Why a figure can have several lines of symmetry · हिंदी में देखें

Open NCERT p. 228One way to think about it

b

1. The blue line is flat and the red half is above it. Each corner gets a partner straight below it, as far below the line as it is above.
2. From its left end on the blue line, the red outline goes up 2, right 2, up 2, right 2, down 2, and then slants down to the blue line 3 squares further right.
3. So below the line, starting from the same point, draw down 2, right 2, down 2, right 2, up 2, and then slant up to the same end point.

In shortThe red outline and its upside-down copy below the blue line, meeting at the two points on the line.

c

1. The blue line slants up to the right at 45°. Reflecting in it swaps 'up' with 'right' (turning the book, as the hint says, makes it an upright line).
2. From its lower point on the blue line, the red staircase goes up 2, right 1, up 2, right 1, up 1, and then right 3 back to the blue line.
3. So on the other side of the line, from the same point, draw right 2, up 1, right 2, up 1, right 1, and then up 3 to the same end point.

In shortA second staircase below the slanted line: right 2, up 1, right 2, up 1, right 1, up 3.

d

1. The blue line is upright. Each corner gets a partner at the same height, as far right of the line as it is left.
2. From the point on the blue line 1 square below its top, the red outline slants to the point 1 square left and 2 squares down, runs 2 squares right across the line, then goes down 1 and left 1 back to the line.
3. So draw the slant from the same top point to the point 1 square right and 2 down (the right end of the flat edge), and from the left end of the flat edge draw down 1 and right 1 back to the line.

In shortA triangle with its top corner on the blue line and a base 2 squares wide, sitting on a rectangle 2 squares wide and 1 square tall.

e

1. The blue line is flat and the red part is below it. Each corner gets a partner straight above it, as far above the line as it is below.
2. Starting on the blue line, the red outline goes 2 left and 2 down, then 3 right and 2 down, then 2 right and 1 up, then 1 left and 3 up, back to the blue line 2 squares right of the start.
3. So above the line, from the same start, draw 2 left and 2 up, then 3 right and 2 up, then 2 right and 1 down, then 1 left and 3 down, to the same end point.

In shortA closed eight-sided shape: the red part below the line and its mirror image above it, meeting at the two points on the line.

f

1. The blue line slants down to the right at 45°. Reflecting in it swaps 'right' with 'down' and 'left' with 'up'.
2. From the upper point where it touches the blue line, the red outline goes up 2, right 2, slants 2 right and 2 down, goes down 2, then left 2 to a second point on the blue line. From there it goes up 1, slants 1 left and 1 up, and goes left 1 back to the start.
3. So on the other side, from the same start, draw left 2, down 2, a slant 2 down and 2 right, right 2, up 2 (to the same second point), then left 1, a slant 1 up and 1 left, and up 1 back to the start.

In shortThe red shape and its mirror image on the other side of the slanted line, touching the line at the same two points.

Watch this explained “Half a figure on squared paper, finished by counting”, 6:38 into Symmetry as reflection: the fold line acts as a mirror · हिंदी में देखें

Open NCERT p. 229One way to think about it

a

1. Reflect the red part in one blue line, reflect it in the other blue line, then reflect one of those copies in the other line: one piece becomes four.
2. Here the blue lines are the two diagonals, crossing at the centre. The red part is a flat segment 4 squares long, 2 squares below the centre, with its ends on the two diagonals.
3. Its reflections are an upright side 2 squares left of the centre, an upright side 2 squares right of it, and a flat side 2 squares above it.

In shortA square 4 squares by 4 squares, centred where the blue lines cross, with the blue lines as its diagonals.

b

1. The blue lines are the two diagonals, crossing at the centre. The red part is an upright edge 2 squares left of the centre, running from one diagonal to the other, with a small pointed bump sticking out to the left at its middle.
2. Its three copies are the same edge on the top, the right and the bottom, each with the bump pointing outwards.

In shortA square 4 squares by 4 squares, centred where the blue lines cross, with a small pointed bump sticking out from the middle of each of its 4 sides.

c

1. The blue lines are upright and flat, crossing at the centre. The red part is in the top-left quarter: from the top of the upright line it goes 1 left, 1 down, 1 left, 1 down, 1 right, 1 down, 1 left and 1 down, reaching the flat line.
2. Copy it into the top-right quarter (mirror in the upright line), the bottom-left quarter (mirror in the flat line) and the bottom-right quarter (mirror of a mirror).

In shortA closed stepped figure made of 4 copies of the red part, one in each quarter, joined on the blue lines.

d

1. The blue lines are upright and flat. The red part is in the bottom-left quarter: from the flat line 3 squares left of the centre, it goes down 2, slants 2 right and 2 down, and then goes 1 right to the upright line.
2. Copy it into the other three quarters.

In shortAn eight-sided figure with corners 3 squares left and right of the centre at 2 squares above and below it, and corners 1 square left and right of the centre at 4 squares above and below it.

e

1. The blue lines are upright and flat. The red part is in the top-right quarter: from the upright line 2 squares above the centre, it slants 1 right and 1 up, goes down 2, right 1, and then down 1 to the flat line.
2. Copy it into the other three quarters.

In shortA closed figure made of the red part and its 3 mirror copies, one in each quarter, meeting on the blue lines 2 squares from the centre.

f

1. The blue lines are upright and flat. The red part starts at the centre and lies in the bottom-left quarter: it slants 3 left and 1 down, goes down 1, slants 1 right and 1 down, and then slants 2 right and 1 up to the upright line.
2. Mirror it in the upright line to close it, then mirror that shape in the flat line.

In shortTwo closed shapes, one below the flat line and one above it, each made of the red part and its mirror image in the upright line. They meet at the centre.

Watch this explained “Two mirror lines want three more copies, not one”, 7:33 into Symmetry as reflection: the fold line acts as a mirror · हिंदी में देखें

Open NCERT p. 230One way to think about it

1. There is more than one correct answer for each figure: the new lines need not close the shape, and a different line of symmetry can be chosen. One answer is shown for each, closing the shape with two lines to one new dot.
2. Number each grid's dots across 0 to 5 from the left and down 0 to 5 from the top, and write a dot as (across, down). Figures 1 to 3 are the top row and 4 to 6 the bottom row, left to right. Use a dot grid with equal spacing across and down.
3. Figure 1: the given lines are (4, 0) to (0, 1) to (4, 5). Add (4, 0) to (5, 1) and (5, 1) to (4, 5). The line of symmetry is the slanted line through (0, 5) and (5, 0).
4. Figure 2: the given lines are (2, 0) to (0, 3) to (1, 5) to (3, 5). Add (3, 5) to (4, 3) and (4, 3) to (2, 0). The line of symmetry is the upright line through (2, 0).
5. Figure 3: the given lines are (0, 0) to (2, 1) to (4, 0) to (4, 2) to (2, 5). Add (2, 5) to (0, 2) and (0, 2) to (0, 0). The line of symmetry is the upright line through (2, 1) and (2, 5).
6. Figure 4: the given lines are (0, 0) to (4, 0) to (4, 1) to (1, 4). Add (1, 4) to (0, 4) and (0, 4) to (0, 0). The line of symmetry is the slanted line through (0, 0) and (5, 5).
7. Figure 5: the given lines are (2, 5) to (0, 1) to (2, 3). Add (2, 3) to (4, 1) and (4, 1) to (2, 5). The line of symmetry is the upright line through (2, 3) and (2, 5).
8. Figure 6: the given lines are (0, 0) to (2, 0) to (5, 2) to (2, 4). Add (2, 4) to (0, 4) and (0, 4) to (0, 0). The line of symmetry is the flat line through (5, 2).

In shortOne answer for each figure (others are possible): 1, add (4, 0) to (5, 1) to (4, 5); 2, add (3, 5) to (4, 3) to (2, 0); 3, add (2, 5) to (0, 2) to (0, 0); 4, add (1, 4) to (0, 4) to (0, 0); 5, add (2, 3) to (4, 1) to (2, 5); 6, add (2, 4) to (0, 4) to (0, 0).

Watch this explained “Half a figure on squared paper, finished by counting”, 6:38 into Symmetry as reflection: the fold line acts as a mirror · हिंदी में देखें

Open NCERT p. 235Matches NCERT’s answer

(a)

1. The figure is a plus sign: four equal arms of equal width, at right angles, meeting at the marked point.
2. A quarter turn (90°) sends each arm onto the next one, so the figure looks the same. So do 180°, 270° and the full turn, 360°.

Answer90°, 180°, 270° and 360°.

(b)

1. The figure is an upright line with a small square at each end, and both squares are on the right-hand side of the line. The marked point is the middle of the line.
2. A half turn sends the top square to the bottom end but onto the left-hand side, where there is no square. So 180° does not work, and no other turn short of 360° works either.

AnswerOnly 360°. The figure has no rotational symmetry.

(c)

1. The figure is an H: two equal upright lines joined by a cross-bar through the marked point.
2. A half turn swaps the two uprights and turns the cross-bar onto itself, so 180° works. A quarter turn would lay the uprights flat, so 90° and 270° do not work.

Answer180° and 360°.

Watch this explained “Reading the angles off figures with no arms at all”, 5:32 into Counting the angles of symmetry of polygons and radial-arm figures · हिंदी में देखें

Open NCERT p. 235Matches NCERT’s answer

1. Number the figures 1 to 7, left to right, top row first. A figure has more than one angle of symmetry when some turn short of 360° works.
2. Figure 1 (a circle with an upright and a flat diameter): a quarter turn works. Angles 90°, 180°, 270°, 360°.
3. Figure 2 (the upside-down triangle): as printed, its top side is shorter than its two slanting sides, so it is not equilateral and only the full turn works. Angle 360° only. (An equilateral triangle would have 120°, 240° and 360°.)
4. Figure 3 (a circle cut by three radii 120° apart): angles 120°, 240°, 360°.
5. Figure 4 (four curved blades at right angles, all curving the same way round): angles 90°, 180°, 270°, 360°.
6. Figure 5 (two equal lines crossing at right angles): angles 90°, 180°, 270°, 360°.
7. Figure 6 (a regular five-pointed star): angles 72°, 144°, 216°, 288°, 360°.
8. Figure 7 (an upright line with two half-circles on the same side): a half turn would put the half-circles on the other side, so only 360° works.

AnswerFigures 1, 3, 4, 5 and 6.

Watch this explained “Five arms, six arms, twelve: the same division again”, 3:47 into Counting the angles of symmetry of polygons and radial-arm figures · हिंदी में देखें

Open NCERT p. 236Checked by computerReads two ways: both answers shown

1. The order of rotational symmetry is the number of times a figure looks exactly the same during one full turn (360°) about the marked point.
2. 3rd figure, six-pointed star: it looks the same after every 60° turn. 360° ÷ 60° = 6, so the order is 6.
3. 4th figure, three arms: the three arms are the same and 120° apart, so it looks the same after 120°, 240° and 360°. Order 3.
4. 5th figure, plus shape: four equal arms, the same after every 90°. Order 4.
5. 6th figure, pentagon: five equal sides, the same after every 72°. Order 5.
6. 1st figure, a line with a flag at each end: a half turn (180°) takes the top flag to the bottom. The figure is meant to have the same flag at each end, so it matches after 180° and 360°: order 2. Look closely at the print, though: the lower flag is drawn flipped (its point is towards the end of the line, while the top flag's point is towards the middle). Taking the drawing exactly, only the full turn works, so the order would be 1.
7. 2nd figure, an X with an arc: the two lines are equal and drawn exactly at right angles, but an arc is drawn in the top angle, and the book does not say what it means. Read as part of the figure: any turn less than 360° moves the arc to another angle, so the order is 1.
8. Read as the two lines only: they look the same after every 90°, order 4. Read as a mark saying the angle is not a right angle: the lines match only after 180° and 360°, order 2.

Answer1st (flags): 2, taking the flags as the same flag (as printed the lower flag is flipped, which strictly gives 1). 2nd (X): with the arc as part of the figure 1; the two lines alone 4; if the arc means the angle is not a right angle 2. 3rd (star): 6. 4th (three arms): 3. 5th (plus): 4. 6th (pentagon): 5.

Watch this explained “Order of rotational symmetry, and a circle of twelve slices”, 9:19 into Counting the angles of symmetry of polygons and radial-arm figures · हिंदी में देखें

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i) 3 angles of symmetry

1. Number the 12 sectors 1 to 12 in order round the circle. The sector lines stay drawn, so only turns of whole sectors (30° each) can work, and a turn works when every coloured sector lands on a coloured sector.
2. For exactly 3 angles, the pattern must repeat every 4 sectors (4 × 30° = 120°) and not more often. Colour sectors 1, 2, 5, 6, 9 and 10, and leave 3, 4, 7, 8, 11 and 12 white.
3. Turns of 120°, 240° and 360° work. A turn of 30°, 60°, 90° or 180° puts a coloured sector onto a white one.

AnswerColour sectors 1, 2, 5, 6, 9, 10: angles of symmetry 120°, 240°, 360° (3 angles).

ii) 4 angles of symmetry

1. For exactly 4 angles, the pattern must repeat every 3 sectors (3 × 30° = 90°) and not more often. Colour sectors 1, 4, 7 and 10.
2. Turns of 90°, 180°, 270° and 360° work. Any other turn moves sector 1 onto a white sector.

AnswerColour sectors 1, 4, 7, 10: angles of symmetry 90°, 180°, 270°, 360° (4 angles).

iii) the possible numbers of angles of symmetry you can obtain

1. Every working turn is a whole number of sectors, and doing a working turn twice still works. So the working turns are the multiples of the smallest one, and that smallest turn must go exactly into 12 sectors. The number of angles is therefore 1, 2, 3, 4, 6 or 12, the numbers that divide 12.
2. Each of these happens: every sector the same colour gives 12; alternate sectors (1, 3, 5, 7, 9, 11) give 6; sectors 1, 4, 7, 10 give 4; sectors 1, 2, 5, 6, 9, 10 give 3; sectors 1 and 7 give 2; sector 1 alone gives 1.

Answer1, 2, 3, 4, 6 or 12 angles of symmetry, the numbers that divide 12.

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Open NCERT p. 238Checked by computerAnswers can differ: one example

1. Many figures work, for example any regular polygon other than the square, a rectangle that is not a square, or a rhombus that is not a square. Here are two.
2. An equilateral triangle has 3 lines of symmetry (each through a corner and the middle of the opposite side) and angles of symmetry 120°, 240° and 360°.
3. A regular hexagon has 6 lines of symmetry and angles of symmetry 60°, 120°, 180°, 240°, 300° and 360°.

AnswerFor example, an equilateral triangle and a regular hexagon. Other answers are possible.

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Open NCERT p. 238Checked by computerAnswers can differ: one example

a at least two lines of symmetry and at least two angles

1. All four are possible, and many shapes work; one sketch is given for each, by its corners.
2. The triangle with corners (0, 0), (4, 0), (2, 2√3) has all sides 4 units long, so it is equilateral.
3. It has 3 lines of symmetry and angles of symmetry 120°, 240° and 360°, so at least two of each.

AnswerAn equilateral triangle, for example (0, 0), (4, 0), (2, 2√3).

b only one line of symmetry but not having rotational symmetry

1. The triangle (0, 0), (4, 0), (2, 3) has two equal sides (√13 each) and a base of 4, so it is isosceles but not equilateral.
2. Its one line of symmetry is the upright line x = 2.
3. Its top corner has a different angle from the other two, so no turn short of 360° can carry it onto another corner. It has no rotational symmetry.

AnswerAn isosceles triangle that is not equilateral, for example (0, 0), (4, 0), (2, 3).

c A quadrilateral with rotational symmetry but no reflection symmetry.

1. The quadrilateral (0, 0), (4, 0), (5, 2), (1, 2) is a parallelogram with sides 4 and √5 and no right angle.
2. A half turn about its centre (2.5, 1) swaps opposite corners and opposite sides, so 180° is an angle of symmetry.
3. It has no line of symmetry, because its neighbouring sides are unequal and its corners are not right angles.

AnswerA parallelogram that is not a rectangle or a rhombus, for example (0, 0), (4, 0), (5, 2), (1, 2).

d A quadrilateral with reflection symmetry but not having rotational symmetry.

1. The quadrilateral (0, 3), (2, 0), (0, −1), (−2, 0) is a kite with sides √13, √5, √5, √13.
2. Its line of symmetry is the upright line x = 0.
3. The top corner is sharper than every other corner, so no turn short of 360° can carry it onto another corner. It has no rotational symmetry.

AnswerA kite that is not a rhombus, for example (0, 3), (2, 0), (0, −1), (−2, 0).

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1. Every angle of symmetry of a figure is a whole-number multiple of its smallest angle of symmetry.
2. So list the multiples of 60 up to 360: 60, 120, 180, 240, 300, 360.
3. The smallest one (60) is already given, so the other angles are the rest of the list.

Answer120°, 180°, 240°, 300° and 360°.

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1. The angles of symmetry are the multiples of the smallest one: the smallest angle, 2 times it, 3 times it, and so on up to 360°.
2. 60° is one of these, and exactly two of them are less than 60°: the smallest angle and twice the smallest angle. So 60° is 3 times the smallest angle.
3. Smallest angle = 60° ÷ 3 = 20°.
4. Check: the angles are 20°, 40°, 60°, 80°, and so on up to 360°. Exactly two of them (20° and 40°) are less than 60°, and 20 goes into 360 exactly 18 times.

Answer20°

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1. The smallest angle of symmetry has to fit into 360 degrees a whole number of times (otherwise adding it to itself would eventually overshoot 360 and leave behind an even smaller working angle).
2. 45 goes into 360 exactly 8 times, so 45 degrees is possible (a figure of order 8).
3. 17 goes into 360 only 21 times with 3 left over, so 17 degrees is not possible for any figure at all.

Answera. Yes (order 8). b. No -- 17 does not divide 360.

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Open NCERT p. 239Matches NCERT’s answer

a Does the outer boundary of the picture have reflection symmetry?

1. The outline is an equilateral triangle with its three corners chamfered off.
2. Measured off the page, the three long sides are all close to equal, and the three short (chamfered) sides are all close to equal to each other, but a long side is clearly not equal to a short side.
3. So the hexagon keeps only the underlying triangle's 3 lines of symmetry, not 6.

AnswerYes -- 3 lines of symmetry, each through the midpoint of one short (chamfered) side and the opposite long side's midpoint.

b Does it have rotational symmetry around its centre?

1. The same alternating long/short side pattern that gives 3 lines of symmetry also gives order-3 rotational symmetry about the centre.

AnswerYes -- 120 degrees, 240 degrees and 360 degrees.

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1. Chapter 1, Table 3 shows eight shapes in the Regular Polygons sequence: triangle, quadrilateral (a square), pentagon, hexagon, heptagon, octagon, nonagon and decagon. These have 3, 4, 5, 6, 7, 8, 9 and 10 equal sides.
2. In a regular polygon all sides are equal and all angles are equal, so the line from any corner through the centre is a line of symmetry.
3. If the number of sides is odd, each such line goes from a corner to the middle of the opposite side, so there is one line for each corner: 3 sides give 3 lines, 5 sides give 5 lines, and so on.
4. If the number of sides is even, each line through a corner also passes through the opposite corner, giving half as many lines as corners. The lines through the midpoints of opposite sides are lines of symmetry too, giving the other half. So again there are as many lines as sides: a square has 2 + 2 = 4.
5. So a regular polygon has as many lines of symmetry as it has sides: triangle 3, square 4, pentagon 5, hexagon 6, heptagon 7, octagon 8, nonagon 9, decagon 10.

AnswerThe lines of symmetry are 3, 4, 5, 6, 7, 8, 9, 10. This is the sequence of counting numbers starting from 3, the same as the number of sides of each polygon.

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1. Chapter 1, Table 3 shows eight shapes in the Regular Polygons sequence: triangle, quadrilateral (a square), pentagon, hexagon, heptagon, octagon, nonagon and decagon. These have 3, 4, 5, 6, 7, 8, 9 and 10 equal sides.
2. An angle of symmetry is a turn about the centre that makes the figure look exactly the same. The book counts the full turn of 360° as one of them, since every figure comes back to itself after a full turn.
3. Turning a regular polygon with n sides by 360° ÷ n moves every corner onto the next corner, so the polygon fits onto itself. Doing this 1, 2, 3, … up to n times gives n angles of symmetry, the last one being 360°.
4. Triangle: 120°, 240°, 360°, so 3 angles. Square: 90°, 180°, 270°, 360°, so 4 angles. Pentagon: 72°, 144°, 216°, 288°, 360°, so 5 angles.
5. In the same way the hexagon (every 60°) has 6, the heptagon has 7, the octagon (every 45°) has 8, the nonagon (every 40°) has 9 and the decagon (every 36°) has 10.

AnswerThe angles of symmetry are 3, 4, 5, 6, 7, 8, 9, 10. This is again the counting numbers starting from 3, the same sequence as the lines of symmetry.

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1. Chapter 1, Table 3 shows five shapes in the Koch Snowflake sequence. The first is an equilateral triangle. Each next shape is made by pushing an equilateral bump outwards from the middle third of every edge.
2. Triangle: 3 lines of symmetry, one through each corner, and 3 angles of symmetry: 120°, 240° and 360°.
3. Second shape, the six-pointed star: 6 lines of symmetry (3 through opposite points of the star and 3 through opposite inner corners) and 6 angles of symmetry: 60°, 120°, 180°, 240°, 300° and 360°.
4. Each later shape is made by doing exactly the same thing to every edge, so every fold and every turn that fits the star onto itself also fits the new shape onto itself. No extra symmetry appears: the six points of the star stay the six farthest points of every later shape, and they can only be matched to one another in these 6 folds and 6 turns.
5. So the third, fourth and fifth shapes also have 6 lines and 6 angles of symmetry.

AnswerLines of symmetry: 3, 6, 6, 6, 6. Angles of symmetry: 3, 6, 6, 6, 6. After the first triangle, every shape in the sequence has 6 lines and 6 angles of symmetry.

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1. The Ashoka Chakra has 24 equal spokes, evenly spaced, so neighbouring spokes are 360° ÷ 24 = 15° apart.
2. Turning it by 15° moves every spoke onto the next one, so the angles of symmetry are 15°, 30°, 45° and so on up to 360°: 24 angles.
3. Lines of symmetry: 12 lines each run along a pair of opposite spokes, and 12 more run exactly halfway between neighbouring spokes. That is 24 lines.

Answer24 lines of symmetry and 24 angles of symmetry (the smallest is 15°).

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