Teach Me Maths

Symmetry

Question 11 mark

Here is a rectangle.

How many lines of symmetry does the rectangle have?

Hint

A line of symmetry folds the shape so that one half fits exactly on top of the other. Think about the diagonals carefully.

Worked solution

Here is a rectangle.

How many lines of symmetry does the rectangle have?

  1. Folding along the vertical line through the middle gives two halves that match.
  2. Folding along the horizontal line through the middle also gives two halves that match.
    Two folds that make the rectangle's halves match Two copies of the rectangle. The first has a vertical fold through its centre and a pair of matching points equally far from the fold, one on each side. The second has a horizontal fold through its centre and a pair of matching points equally far above and below it. Matching distance marks show why the points meet when folded. In each copy, the two halves are shaded differently. Vertical fold Horizontal fold Matching points meet when folded. The marks show equal distances.
    The dotted guides meet each fold at a right angle. Matching marks show equal distances, so the paired points meet when folded.
  3. A diagonal is not a line of symmetry: the two halves do not fit on top of each other when folded.
    Why a diagonal fold does not work A rectangle 240 drawing units wide and 120 high has a diagonal fold from its bottom left corner to its top right corner. The bottom right triangular half is pale blue. The top left triangular half is shown after folding, in orange. Its top left corner lands below the rectangle, so the two halves do not coincide. A dotted segment joins that corner to its folded position perpendicular to the diagonal. The original rectangle boundary remains visible. Try a diagonal fold A A after folding This corner lands outside the rectangle.
    The orange triangle shows the half after folding. Corner A lands outside the original rectangle, so this fold does not make the halves match.
  4. The rectangle has 2 lines of symmetry.

Question 21 mark

Here is a kite with three dashed lines, A, B and C.

Which dashed line is a line of symmetry of the kite?

Hint

Imagine folding the kite along each line. Do the two halves fit exactly on top of each other?

Worked solution

Here is a kite with three dashed lines, A, B and C.

Which dashed line is a line of symmetry of the kite?

  1. Line A
  2. Line B
  3. Line C
  1. Line B goes through the top and bottom corners of the kite. Each side of it is a reflection of the other, so B is a line of symmetry.
    Line B folds the kite onto itselfThe vertical line B is highlighted as a fold. The left and right corners have matching distance marks to B and meet when folded. Lines A and C do not make matching halves.ABC
    The matching marks show equal distances to B, so the two corners meet when folded.
  2. Line A cuts the kite into a small top part and a large bottom part, so the halves do not match.
  3. Line C is slanted, and the two parts it makes are different shapes.
  4. The answer is line B.

Question 31 mark

Here is a parallelogram.

Write down the order of rotational symmetry of the parallelogram.

Hint

Count how many times the shape looks exactly the same as it turns through one full turn.

Worked solution

Here is a parallelogram.

Write down the order of rotational symmetry of the parallelogram.

  1. Turn the parallelogram about its centre.
  2. After a half turn it fits exactly on top of itself. After a full turn it is back where it started.
    A parallelogram after a half turnDashed diagonals meet at the centre of the parallelogram. A 180 degree rotation moves corner A to the opposite corner. Every corner moves to its opposite, so the whole shape fits on itself.AA after 180°180°Turn about the centre
    The marked centre is where the diagonals meet. A moves to the opposite corner after 180 degrees.
  3. So it fits on itself 2 times in a full turn.
  4. The order of rotational symmetry is 2.

Question 42 marks

Here is an isosceles trapezium.

(a)

Write down the number of lines of symmetry of the trapezium.

1 mark

(b)

Write down the order of rotational symmetry of the trapezium.

1 mark

Hint

For rotational symmetry, count how many times the shape looks the same in one full turn, including the end of the turn.

Worked solution

Here is an isosceles trapezium.

Part (a)

Write down the number of lines of symmetry of the trapezium.

  1. The vertical line through the middle of the two parallel sides folds the trapezium onto itself.
    The trapezium has one mirrorA vertical dashed mirror passes through the midpoints of the two parallel sides. Matching points on the sloping sides are equally far from it.
    The marked points meet when folded on the vertical dashed line.
  2. No other fold works, so there is 1 line of symmetry.

Part (b)

Write down the order of rotational symmetry of the trapezium.

  1. The trapezium only looks the same again after a full turn.
    A half turn does not fitThe original trapezium is blue. Its dashed orange outline after a half turn has the long side at the top instead of at the bottom, so the outlines do not coincide.180°
    The orange dashed outline shows a failed half turn: it puts the long side at the top.
  2. It fits on itself just once in a full turn, so the order of rotational symmetry is 1.

Question 51 mark

Here is a pattern of squares on a grid. Five squares are shaded.

One more square is to be shaded so that the dashed line is a line of symmetry of the pattern.

Which square should be shaded?

Hint

Each shaded square needs a matching shaded square the same distance away on the other side of the dashed line.

Worked solution

Here is a pattern of squares on a grid. Five squares are shaded.

One more square is to be shaded so that the dashed line is a line of symmetry of the pattern.

Which square should be shaded?

  1. A
  2. B
  3. C
  4. D
  1. The two shaded squares on each side next to the dashed line already match each other.
  2. The shaded square in the top left corner has no partner yet. Its reflection in the dashed line is the square in the top right corner.
    Reflect the unpaired squareThe original five squares are blue. The added top right square C is orange and hatched. A horizontal guide links the centres of the top left and top right squares, with matching distances to the vertical mirror.CAdd C: top rightSame row, same distance from mirror
    The matching marks show equal perpendicular distances to the mirror. Only the orange hatched square C is added.
  3. That is square C.

Question 61 mark

Here is a regular pentagon.

How many lines of symmetry does the regular pentagon have?

Hint

In a regular polygon all the sides are equal and all the angles are equal. Try a fold from each corner to the middle of the opposite side.

Worked solution

Here is a regular pentagon.

How many lines of symmetry does the regular pentagon have?

  1. Each corner has a fold through it and the middle of the opposite side. For example, folding on the vertical line brings the two marked corners together.
    Five folds of a regular pentagonAll five dashed lines run from a vertex to the midpoint of the opposite side. Each folds the regular pentagon onto itself. The highlighted left and right vertices meet on the vertical fold.
    The five dashed segments show all five lines of symmetry through the shape.
  2. There are 5 corners and one such line through each, so there are 5 lines of symmetry.

Question 71 mark

Here is a regular hexagon.

Write down the order of rotational symmetry of the regular hexagon.

Hint

Count how many times the hexagon looks exactly the same as it makes one full turn about its centre.

Worked solution

Here is a regular hexagon.

Write down the order of rotational symmetry of the regular hexagon.

  1. Turn about the marked centre. A turn of 60° moves each corner onto the next, so the whole hexagon fits exactly on itself.
    A sixty-degree turn moves to the next cornerThe centre is marked. Two radii run to adjacent corners and an arrow shows a 60 degree rotation between them. Each corner then lands on the next corner. The hexagon has two vertical sides.60°
    The arrow shows one of the six equal turns in a full turn.
  2. The matching positions are 60°,120°,180°,240°,300° and 360°.
  3. 360÷60=6, so the order of rotational symmetry is 6.

Question 81 mark

Here are four shapes, A, B, C and D.

Which shape has no lines of symmetry?

Hint

For each shape, imagine folding it so that one half fits exactly on top of the other. Is there any fold that works?

Worked solution

Here are four shapes, A, B, C and D.

Which shape has no lines of symmetry?

  1. A
  2. B
  3. C
  4. D
  1. Try folds that make the two halves fit exactly. The dashed lines show the folds that work for A, B and C.
    Compare possible foldsA has all four symmetry axes drawn, B has its vertical axis, and C has its vertical axis. D has none because this sloping parallelogram is neither a rectangle nor a rhombus.ABCD
    A has 4 lines; B and C have 1 each. D has no line of symmetry.
  2. D is a sloping parallelogram with unequal adjacent sides. A vertical or horizontal fold fails because its sloping sides do not match; a diagonal fold fails too.
  3. The answer is D.

Question 92 marks

Here is the capital letter H.

(a)

Write down the number of lines of symmetry of the letter H.

1 mark

(b)

Write down the order of rotational symmetry of the letter H.

1 mark

Hint

Try folding the letter both down the middle and across the middle. For rotational symmetry, think about turning it upside down.

Worked solution

Here is the capital letter H.

Part (a)

Write down the number of lines of symmetry of the letter H.

  1. The vertical fold swaps the left and right bars. The horizontal fold swaps the top and bottom halves. Both folds match exactly.
    Two symmetry axes of HA vertical and a horizontal dashed line pass through the centre of the H. Folding along either line puts matching bars onto each other.
    Both dashed lines are lines of symmetry.
  2. There are 2 lines of symmetry.

Part (b)

Write down the order of rotational symmetry of the letter H.

  1. Turn the H through 180° about its centre. The top left bar maps to the bottom right, and the top right maps to the bottom left, so the letter fits on itself.
    Half-turn symmetry of HThe centre of the H is marked and a curved arrow shows a half turn about it. Left top maps to right bottom, and right top maps to left bottom, leaving the same outline.180°Centre of rotation
    A half turn leaves exactly the same outline.
  2. It matches at 180° and 360°: 2 positions in a full turn. Its rotational symmetry has order 2.

Question 102 marks

The diagram shows half of a shape drawn on a grid. The dashed line is a line of symmetry of the whole shape.

Point P is a corner of the shape.

Write down the coordinates of the corner that matches P on the other half of the shape.

Write your answer as (x, y)

Hint

The matching corner is the same distance from the dashed line as P, but on the other side.

Worked solution

The diagram shows half of a shape drawn on a grid. The dashed line is a line of symmetry of the whole shape.

Point P is a corner of the shape.

Write down the coordinates of the corner that matches P on the other half of the shape.

  1. The dashed line is the line x=4.
  2. P is at (2,4), which is 2 units to the left of the dashed line.
  3. The matching corner is 2 units to the right of the dashed line, at the same height.
    Locating the mirror image of PAn equal-scale coordinate grid shows the original half outline joining (4,1), (1,1), P(2,4) and (4,7). The mirror is x=4. A horizontal construction at y=4 is perpendicular to the mirror. P is two units left and its image P prime is two units right, at (6,4). Matching ticks show equal distances. The image keeps the same height.012345678912345678xyMirror: x = 42units2unitsPP′P (2, 4) → P′ (6, 4)Same height: y = 4
    Draw a horizontal guide through P, at right angles to the mirror. Count 2 units to the mirror, then 2 units beyond it. The height stays 4.
  4. 4+2=6, so the corner is at (6,4).

Question 111 mark

A quadrilateral has no lines of symmetry. It has rotational symmetry of order 2.

What type of quadrilateral is it?

Hint

Think about which quadrilaterals look the same after a half turn. Then rule out any that have a line of symmetry.

Worked solution

A quadrilateral has no lines of symmetry. It has rotational symmetry of order 2.

What type of quadrilateral is it?

  1. Rectangle
  2. Rhombus
  3. Parallelogram
  4. Kite
  1. A rectangle has mirror lines through the middles of opposite sides; a rhombus has mirror lines along its diagonals. A square has 4 mirror lines, so these all fail the condition of no lines of symmetry.
  2. A kite also has a line of symmetry, so it fails that condition.
  3. A parallelogram that is neither a rectangle nor a rhombus has no lines of symmetry. A half turn about its centre swaps opposite corners, giving rotational symmetry of order 2.
    A parallelogram after a half turnDashed diagonals meet at the centre of the parallelogram. A 180 degree rotation moves corner A to the opposite corner. Every corner moves to its opposite, so the whole shape fits on itself.AA after 180°180°Turn about the centre
    Opposite corners swap after 180 degrees; the sloping shape has no matching fold.
  4. Choose parallelogram: the conditions rule out its special cases, rectangles and rhombuses.

Question 122 marks

Here is a pattern of squares on a grid. Five squares are shaded.

One more square is to be shaded so that the pattern has rotational symmetry of order 2.

Which square should be shaded?

Hint

Turn the pattern through half a turn about the centre of the grid. Which shaded square does not land on another shaded square?

Worked solution

Here is a pattern of squares on a grid. Five squares are shaded.

One more square is to be shaded so that the pattern has rotational symmetry of order 2.

Which square should be shaded?

  1. A
  2. B
  3. C
  4. D
  1. Use a half turn of 180° about the centre of the grid. Each shaded square needs a partner directly opposite the centre.
  2. The two pairs numbered 1 and 2 already match. The square numbered 3 in row 2, column 3 needs its partner in row 3, column 2: square C.
    Complete the half-turn pairsThe grid centre is marked. A 180 degree turn swaps equal-numbered squares. Pairs 1 and 2 are already shaded blue. In pair 3, the second-row third square maps to C, the third-row second square, now hatched orange. The dotted line passes through the centre.123321180°Pairs 1 and 2 already match.Add C to complete pair 3.
    Equal numbers show squares that swap in a half turn. Only C is added.
  3. Shade C. All three pairs now match after a half turn, so the answer is C.

Question 131 mark

A logo has rotational symmetry of order 8.

Work out the smallest angle the logo must turn through about its centre so that it looks exactly the same.

Hint

The logo looks the same 8 times during one full turn, and these turns are all the same size.

Worked solution

A logo has rotational symmetry of order 8.

Work out the smallest angle the logo must turn through about its centre so that it looks exactly the same.

  1. One full turn is 360°. Order 8 means there are 8 equally spaced matching positions in that turn, including the full-turn position.
  2. The smallest matching turn is one of these 8 equal intervals: 360÷8=45°.

Question 142 marks

ABCD is a kite. Points A, B and C are shown on the grid.

The dashed line is a line of symmetry of the kite.

Write down the coordinates of point D.

Write your answer as (x, y)

Hint

D is the reflection of B in the dashed line.

Worked solution

ABCD is a kite. Points A, B and C are shown on the grid.

The dashed line is a line of symmetry of the kite.

Write down the coordinates of point D.

  1. The mirror is y=3. A and C lie on it, so they stay in place.
  2. B is at (3,5), which is 5-3=2 units above the mirror. Put D directly below B, the same 2 units below the mirror.
    Reflect B across y = 3The completed kite has B at (3,5) and D at (3,1). A vertical dotted guide joins them, perpendicular to y = 3. Matching marks and labels show two units above and two units below the mirror. A and C stay on the mirror.012345678910123456xyy = 32 units2 unitsAB (3,5)CD (3,1)
    The guide is perpendicular to the mirror. The two marked distances are equal.
  3. The x-coordinate stays 3. The new y-coordinate is 3-2=1, so D is at (3,1).

Question 152 marks

The diagram shows a hexagon. The dashed line is a line of symmetry of the hexagon.

The perimeter of the hexagon is 38 cm.

Work out the length of the bottom side, x.

Hint

Use the line of symmetry to find the lengths of the sides on the right-hand side first.

Worked solution

The diagram shows a hexagon. The dashed line is a line of symmetry of the hexagon.

The perimeter of the hexagon is 38 cm.

Work out the length of the bottom side, x.

  1. The mirror pairs the two upper sloping sides and the two lower sloping sides. Copy 5 cm and 7 cm onto the right.
    Use symmetry to label all six sidesThe upper pair of sloping sides each have one matching tick and are 5 cm. The lower pair each have two matching ticks and are 7 cm. The top is 6 cm and the bottom is x cm.6 cm5 cm7 cmx cmDiagram not to scale5 cm7 cm
    One tick pairs the 5 cm sides; two ticks pair the 7 cm sides.
  2. The perimeter includes all six sides. The five known lengths total 6+5+5+7+7=30 cm.
  3. The bottom side is what is left: x=38-30=8 cm.

Question 16Challenge4 marks

Sam bends a piece of wire 42 cm long to make this pentagon. He uses all of the wire.

The dashed line is a line of symmetry of the pentagon. Three of the side lengths are shown, in centimetres.

(a)

Work out the value of x.

3 marks

(b)

Work out the length of one of the sloping sides of the pentagon.

1 mark

Hint

Use the line of symmetry to write down the lengths of the two unlabelled sides. The five sides add up to the length of the wire.

Worked solution

Sam bends a piece of wire 42 cm long to make this pentagon. He uses all of the wire.

The dashed line is a line of symmetry of the pentagon. Three of the side lengths are shown, in centimetres.

Part (a)

Work out the value of x.

  1. Symmetry gives two upright sides of x+4 and two sloping sides of 2x-1. The bottom side is 3x.
    Label both pairs of matching sidesThe symmetric house-shaped pentagon has base 3x. Both upright sides are x plus 4, with one tick each. Both sloping sides are 2x minus 1, with two ticks each. The two right-hand labels have been added in orange.3xx + 4x + 42x − 12x − 1Matching sides have matching marks.Diagram not to scale
    Use each of the five sides exactly once in the perimeter.
  2. All 42 cm of wire forms the perimeter, so 3x+2(x+4)+2(2x-1)=42.
  3. Collect like terms: 3x+2x+8+4x-2=42, so 9x+6=42.
  4. Subtract 6: 9x=36. Divide by 9: x=4.

Part (b)

Work out the length of one of the sloping sides of the pentagon.

  1. Each sloping side is 2x-1 cm. Substitute x=4: 2\times4-1=7 cm.
  2. Check: the five sides are 12,8,8,7,7 cm, which total 42 cm.

Question 17Challenge4 marks

The diagram shows a hexagon ABCDEF. The dashed line through A and D is a line of symmetry of the hexagon.

Angle FAB =110° and angle ABC =125°.

Angle CDE is 30° more than angle BCD.

Work out the size of angle CDE.

Hint

The angles in a hexagon add up to 720°. Use the line of symmetry to find which angles are equal.

Worked solution

The diagram shows a hexagon ABCDEF. The dashed line through A and D is a line of symmetry of the hexagon.

Angle FAB =110° and angle ABC =125°.

Angle CDE is 30° more than angle BCD.

Work out the size of angle CDE.

  1. The mirror maps B to F and C to E. So angle AFE is 125°, and angles BCD and DEF are equal. Call each of these y°; angle CDE is then (y+30)°.
    Pair the angles before calculatingThe mirror through A and D pairs B with F, both 125 degrees, and C with E, both labelled y. D is y plus 30 degrees. A is 110 degrees. Matching single arcs mark B and F and matching double arcs mark C and E.ABCDEF110°125°125°yyy + 30°
    Match the angles across the mirror before writing the equation.
  2. A hexagon has angle sum (6-2)\times180=720°.
  3. The known angles total 110+125+125=360°, leaving 720-360=360° for C, D and E.
  4. So y+(y+30)+y=360. Then 3y+30=360, 3y=330 and y=110.
  5. The question asks for angle CDE, not y: 110+30=140°.

Question 18Challenge3 marks

Here is a pattern of squares on a grid. Seven squares are shaded.

More squares are to be shaded so that both dashed lines are lines of symmetry of the pattern.

What is the smallest number of extra squares that must be shaded?

Hint

Reflect every shaded square in the vertical dashed line and in the horizontal dashed line. Count the squares that are needed but not yet shaded.

Worked solution

Here is a pattern of squares on a grid. Seven squares are shaded.

More squares are to be shaded so that both dashed lines are lines of symmetry of the pattern.

What is the smallest number of extra squares that must be shaded?

  1. Every shaded square needs a partner across each dashed line. Work through three groups, adding only partners that are missing.
  2. The bottom-left corner is the missing partner of both the top-left and bottom-right corners. Add this square once: 1 new square.
    1. Match the cornersA six-column, four-row grid with both central dashed mirror axes. The seven original cells remain blue. Orange hatched cells numbered 1 to 1 are the cumulative additions. Green arrows reflect the top-left and bottom-right original corner cells to the same bottom-left cell. Equal-distance ticks and right-angle marks identify the perpendicular mirror constructions. Add that cell once.1. Match the cornersTwo dashed mirror lines1OriginalAdded1 new squareSame square: count it once.
    The two arrows lead to the same missing square. The hatched square numbered 1 is the only addition at this stage.
  3. Reflect the shaded pair in the second row across the horizontal line. Add their partners in the third row: 2 new squares.
    2. Reflect the next pairA six-column, four-row grid with both central dashed mirror axes. The seven original cells remain blue. Orange hatched cells numbered 1 to 3 are the cumulative additions. Green arrows cross the horizontal mirror perpendicularly from row two columns two and five to row three columns two and five. Matching distance marks show equal distances either side. These two cells are additions two and three.2. Reflect the next pairTwo dashed mirror lines123OriginalAdded2 more: 3 added so farKeep the first added square.
    Keep square 1 and add squares 2 and 3. Each new square is the same distance from the horizontal mirror as its partner.
  4. Reflect the middle shaded pair in the third row across the horizontal line. Add their partners in the second row: 2 new squares.
    3. Complete the last pairA six-column, four-row grid with both central dashed mirror axes. The seven original cells remain blue. Orange hatched cells numbered 1 to 5 are the cumulative additions. Green arrows cross the horizontal mirror perpendicularly from row three columns three and four to row two columns three and four. These are additions four and five. The completed pattern matches across both mirrors.3. Complete the last pairTwo dashed mirror lines12345OriginalAdded2 more: 5 added in totalBoth mirrors now match.
    Add squares 4 and 5. The completed pattern now matches across both dashed lines.
  5. Every shaded square now has both mirror partners. Only the five numbered squares were added: 1+2+2=5 extra squares.

Question 19Challenge2 marks

The kite in this question is not a rhombus.

Kim says,

"A kite has rotational symmetry of order 2, because it has two pairs of equal sides."

Ali says,

"A kite has the same number of lines of symmetry as an isosceles trapezium."

Who is right?

Hint

Picture a kite turning through half a turn. Then count the lines of symmetry of each shape.

Worked solution

The kite in this question is not a rhombus.

Kim says,

"A kite has rotational symmetry of order 2, because it has two pairs of equal sides."

Ali says,

"A kite has the same number of lines of symmetry as an isosceles trapezium."

Who is right?

  1. Only Kim
  2. Only Ali
  3. Both Kim and Ali
  4. Neither of them
  1. The kite has one mirror line through its top and bottom corners. But the half-turn image does not fit: the short and long sections swap places.
    Compare a kite and an isosceles trapeziumTop: a non-rhombus kite has one vertical mirror, with equally distant left and right corners. Middle: rotating the kite 180 degrees about its centre produces a dashed orange outline that does not fit. Bottom: an isosceles trapezium also has one vertical mirror.Kite: one foldKite: test a half turn180°Dashed outline does not fit.Isosceles trapezium: one foldBoth have exactly one mirror line.
    The dashed orange half-turn outline fails to match. Each shape has one vertical mirror.
  2. The kite only matches again after a full turn, so its rotational symmetry has order 1. Kim is wrong.
  3. The kite and the isosceles trapezium each have 1 line of symmetry. Ali is right.
  4. The answer is only Ali.

Question 20Challenge4 marks

The diagram shows a shape made from two rectangles. The dashed line is a line of symmetry of the shape.

The area of the shape is 76 cm².

Work out the value of h.

Hint

Use the line of symmetry to work out the width of the lower rectangle first.

Worked solution

The diagram shows a shape made from two rectangles. The dashed line is a line of symmetry of the shape.

The area of the shape is 76 cm².

Work out the value of h.

  1. The mirror makes the two overhangs equal, so the right overhang is also 3.5 cm. The lower rectangle is 12-3.5-3.5=5 cm wide.
    Split the T shape into two rectanglesThe top rectangle is blue and has area 36 square centimetres. The lower rectangle is orange and has area 40 square centimetres. Both overhangs are 3.5 centimetres, so the lower rectangle is 5 centimetres wide. Its height is h.12 cm3 cm3.5 cmh cmDiagram not to scale3.5 cm36 cm²40 cm²5 cm
    Split along the dotted horizontal line into two rectangles that do not overlap.
  2. The top rectangle has area 12\times3=36 cm².
  3. The lower rectangle has the remaining area: 76-36=40 cm².
  4. For the lower rectangle, width \times height = area: 5h=40. So h=40÷5=8.

More on this topic: Symmetry worksheet with full solutions

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