Geometry and Measurement · Position and Transformation · Lesson 1

Lines of Symmetry in Adinkra and Everyday Designs

Look at a kente strip, an Adinkra stamp, the Ghana flag or the letter H: they all look the same on both sides of an invisible middle line. In this lesson you will learn to find that line, count the lines in different shapes and use symmetry to understand reflection, which you will draw on a grid in the next lessons.

NaCCA B7.3.3.1.1 ⏱ about 40 min

By the end you will be able to

✔Explain reflectional (fold) symmetry and a line of symmetry in your own words.
✔State the number of lines of symmetry of common plane shapes, including triangles, quadrilaterals and regular polygons.
✔Identify objects and designs in real life, such as Adinkra symbols, letters and flags, that have reflectional symmetry.
✔Find in how many ways one more square can be added to a shape so that it has a line of symmetry.

Key words

Line of symmetry: A line (also called a fold line or mirror line) that divides a shape into two halves that match exactly when the shape is folded along it.
Reflectional symmetry: A shape has reflectional symmetry if it has at least one line of symmetry. Also called fold or line symmetry.
Mirror image: The picture you see in a mirror: same size and shape as the object, but the left and right sides are swapped.
Congruent: Exactly the same shape and size. The two halves of a symmetrical shape are congruent.

Curriculum reference

Content standard: B7.3.3.1 Perform a single transformation (i.e. reflection and translation) on a 2D shape using graph paper (including technology) and describe the properties of the image under the transformation (i.e. congruence, similarity, etc.)

Indicator: Determine shapes in real life that have reflectional (or fold) symmetries.

Starter

The cloth that folds perfectly

Auntie Efua holds up a piece of Adinkra cloth at the Ntonso market near Kumasi. She folds it down the middle and the stamped patterns on the left land exactly on the patterns on the right, not a thread out of place. 'That is why people say a good design is balanced,' she smiles. Can you find the fold that makes a shape match itself? In this lesson you will become a pattern detective.

1. What is a line of symmetry?

Take a sheet of paper, draw half of a heart on one side of a crease, and cut along the line while the paper is folded. When you open it, the heart is perfectly balanced. The crease you folded along is the line of symmetry. A shape with at least one such line has reflectional symmetry, also called fold symmetry or line symmetry.

The word reflection comes from mirrors. If you stand a small mirror on the line, the half you can see plus its reflection makes the whole shape again. That is why a line of symmetry is also called a mirror line.

▶ Watch the matching points on each side of the fold line. Notice that they are the same distance from the line.

Every point on one side of the line has a matching point on the other side, and the two points are the same distance from the line. The line joining them meets the mirror line at a right angle (90°). Because of this, the two halves are congruent: the same size and shape, but facing opposite ways, like your left and right hands.

2. Lines of symmetry in plane shapes

Different shapes have different numbers of lines of symmetry. The table gives the ones you must know for the BECE. For a regular polygon (all sides and all angles equal) the number of lines of symmetry is always equal to the number of sides.

Lines of symmetry of common shapes
Shape	Lines of symmetry	Remark
Equilateral triangle	3	Corner to middle of opposite side
Isosceles triangle	1	Through the apex and the middle of the base
Scalene triangle	0	All sides different
Square	4	2 through the sides, 2 diagonals
Rectangle (not a square)	2	Diagonals are not fold lines
Rhombus (not a square)	2	The two diagonals
Parallelogram (not a rectangle or rhombus)	0	Has only half-turn symmetry
Kite	1	The diagonal joining the equal sides
Isosceles trapezium	1	Through the middles of the parallel sides
Regular pentagon	5	Regular polygon: one line per side
Regular hexagon	6	Regular polygon: one line per side
Circle	Infinitely many	Every diameter is a line of symmetry

▶ Compare the fold lines of each shape. Notice that the parallelogram has none.

Letters are good practice too. A, M, T, U, V, W, Y have a vertical line of symmetry. B, C, D, E, K have a horizontal line. H, I, O, X have both a vertical and a horizontal line. F, G, J, L, P, Q, R, S and Z have none, so the letters in the word HIDE are all symmetrical about a horizontal line.

3. Symmetry in Ghanaian designs

Ghanaian art is full of reflectional symmetry. Adinkra symbols printed at Ntonso and woven kente strips often have a balanced centre line. The Akoma (heart) symbol of love and patience has one vertical line of symmetry. The Dwennimmen (ram's horns), a symbol of strength together with humility, is built so that the left horn mirrors the right horn.

Other objects around you: the wings of a butterfly, the front of a trotro, a calabash seen from the front, a school badge, a pair of spectacles, the Black Star on the national football shirt. Many designs are made by drawing half and reflecting it to make the other half; that is exactly what you will do with coordinates.

▶ A kite reflected in its own line of symmetry x = 2 lands exactly on itself: the image B′ is where D was and D′ is where B was.

This is the key link: a line is a line of symmetry exactly when reflecting the shape in that line gives back the same shape. On the grid above the line is x = 2, and each corner is sent to the matching corner, so the kite does not seem to move at all.

4. Making a shape symmetrical

The NaCCA curriculum asks a puzzle: in how many different ways can one more square be shaded in a shape so that it has a line of symmetry? To solve it, try every square that touches the shape along a full side, one at a time, and use the fold test on each new shape.

1Draw the shape on squared paper and list every empty square that shares a full side with it.
2Add one candidate square at a time.
3Test the new shape for a line of symmetry: vertical, horizontal or diagonal.
4Count the candidates that pass the test.

A line of symmetry on a grid can be vertical, horizontal or diagonal (at 45°). On a coordinate plane the vertical and horizontal lines are written as $x = k$ and $y = k$ , and the two diagonal lines through the origin as $y = x$ and $y = - x$ .

▶ Reflect the triangle in the y-axis. The object and its image would match if you folded along the y-axis.

Worked examples

Try each one yourself first, then open the solution and compare every step.

Example 1

Counting fold lines of a square

How many lines of symmetry does a square tile have? Describe them.

Show the solution

Step 1A square has 4 equal sides and 4 right angles.
Step 2Fold through the middles of opposite sides: a vertical line and a horizontal line. That is 2 lines.
Step 3Fold corner to corner: two diagonals. That is 2 more lines.
Step 4Total = 2 + 2 = 4. A regular polygon with 4 sides has 4 lines ✓

Answer: 4 lines of symmetry

Example 2

Is the diagonal a fold line?

Ama says the diagonal of her 6 cm by 4 cm rectangular piece of kente is a line of symmetry. Is she right?

Show the solution

Step 1Fold the rectangle along a diagonal.
Step 2The two corners at the ends of the other diagonal do not meet, because the sides are not all equal (6 cm and 4 cm).
Step 3So the halves do not match.
Step 4The rectangle only has 2 lines: through the middles of the 6 cm sides and through the middles of the 4 cm sides.

Answer: No. A rectangle that is not a square has only 2 lines of symmetry.

Example 3

Letters of a word

Which letters of the word KOFI HAD have a vertical line of symmetry?

Show the solution

Step 1Test each letter with a vertical fold.
Step 2K: left and right differ, no. O: yes. F: no. I: yes. H: yes. A: yes. D: no (D is balanced top and bottom, not left and right).
Step 3Letters with a vertical line: O, I, H, A.

Answer: O, I, H and A

Example 4

Adding one square

Three squares form an L shape: the squares at positions (0,0), (1,0) and (0,1). In how many different ways can one more square, sharing a full side with the shape, be shaded so that the new shape has a line of symmetry?

Show the solution

Step 1Squares touching the L along a side: (−1,0), (0,−1), (2,0), (1,1), (1,−1), (−1,1), (0,2). That is 7 candidates.
Step 2(1,1) gives a 2 by 2 square: 4 lines of symmetry ✓.
Step 3(−1,0) gives a T-shape: the vertical line through the middle works ✓. (0,−1) gives a T-shape on its side: the horizontal line works ✓.
Step 4(2,0) and (0,2) give an L of 4 squares; (1,−1) and (−1,1) give a zig-zag. None of these has a line of symmetry ✗.
Step 5So 3 candidates work.

Answer: 3 ways

Example 5

Matching corners on a grid

A shape is symmetrical about the y-axis. One of its corners is P(4, −3). Write the coordinates of the matching corner.

▶ Reflecting in the y-axis changes the sign of each x-coordinate and keeps the y-coordinate.

Show the solution

Step 1The mirror line is the y-axis (x = 0).
Step 2The matching corner is on the other side of the y-axis and the same distance from it, so the x-coordinate changes sign: 4 → −4.
Step 3It is at the same height, so the y-coordinate stays −3.
Step 4Check: the midpoint of (4, −3) and (−4, −3) is (0, −3), which is on the y-axis ✓.

Answer: (−4, −3)

Summary

A line of symmetry (fold or mirror line) divides a shape into two congruent halves that match exactly when folded.
Equilateral triangle 3, square 4, rectangle 2, rhombus 2, kite 1, isosceles triangle 1, parallelogram 0, circle infinite. A regular polygon has as many lines as sides.
Diagonals of a rectangle or parallelogram are not lines of symmetry: always do the fold test.
Adinkra symbols, kente patterns, letters and flags are everyday examples of reflectional symmetry.
Matching points are the same distance from the mirror line. Reflecting a symmetrical shape in its own line gives the same shape back.

The cloth that folds perfectly

1. What is a line of symmetry?

2. Lines of symmetry in plane shapes

3. Symmetry in Ghanaian designs

4. Making a shape symmetrical

Worked examples

Counting fold lines of a square

Is the diagonal a fold line?

Letters of a word

Adding one square

Matching corners on a grid

Summary

Practice: 15 questions