Draw or count mirror lines by checking whether the figure, punched holes or completed half coincides with its reflection across the line.
1. The axes of symmetry are: (a) vertical line through the centre; (b) diagonal from bottom-left to top-right; (c) horizontal line through the centre; (d) horizontal line through the centre; (e) vertical, horizontal and both diagonal lines through the centre; (f) vertical line through the centre; (g) vertical line through the apex and midpoint of the base; (h) horizontal line through the middle; (i) vertical line through the apex and midpoint of the base; (j) vertical line through the centre; (k) vertical and horizontal lines through the centre; (l) vertical line through the centre.
2. The other hole(s) are the reflections of the given holes in the dotted line: (a) no new hole is needed because the given hole lies on the mirror line; (b) add a hole the same distance above the horizontal line on the right; (c) add a matching hole on the right of the vertical line; (d) add a matching hole on the opposite side of the slant line; (e) add a matching hole on the opposite side of the slant line.
3. The completed figures are: (a) square (b) triangle (c) rhombus (d) circle (e) pentagon (f) octagon.
4. The multiple lines of symmetry are: (a) 3 lines through the centre; (b) vertical and horizontal lines; (c) 3 lines through the vertices and the centre; (d) the two diagonals; (e) vertical, horizontal and both diagonals; (f) only one line of symmetry, the vertical line, so no multiple lines; (g) vertical, horizontal and both diagonals; (h) 6 lines through opposite petals/centres.
5. Shade the mirror images of the four given shaded squares in the chosen diagonal. Yes, there is more than one way, because either diagonal can be chosen. The figure will be symmetric about both diagonals only if all squares needed by both diagonal reflections are shaded.
6. Complete each diagram by drawing the mirror image of the given part in every dotted mirror line: (a) reflect the polygon across the diagonal; (b) reflect the stepped shape first across the vertical line and then across the horizontal line so all four parts match; (c) reflect the curve across the vertical and horizontal lines so the completed curve has both symmetries.
7. (a) 3 (b) 1 (c) 0 (d) 4 (e) 2 (f) 2 (g) 0 (h) 0 (i) 6 (j) infinitely many.
8. (a) A, H, I, M, O, T, U, V, W, X, Y (b) B, C, D, E, H, I, O, X (c) O, X, I, H.
9. Examples: a scalene triangle, a parallelogram that is not a rectangle or rhombus, and an irregular quadrilateral.
10. (a) Median (b) Diameter.
Use the centre of rotation and count how many times the figure matches itself in one complete turn; the basic angle is $360^\circ$ divided by the order.
1. Two examples are a square and an equilateral triangle.
2. Rough sketches: (i) an equilateral triangle; (ii) an isosceles triangle which is not equilateral; (iii) a parallelogram which is not a rectangle or a rhombus; (iv) a kite.
3. Yes.
4. Square: centre at intersection of diagonals, order 4, angle $90^\circ$; Rectangle: centre at intersection of diagonals, order 2, angle $180^\circ$; Rhombus: centre at intersection of diagonals, order 2, angle $180^\circ$; Equilateral Triangle: centre at the common point of medians, order 3, angle $120^\circ$; Regular Hexagon: centre, order 6, angle $60^\circ$; Circle: centre, infinitely many rotations, any angle; Semi-circle: centre of the full circle, order 1, angle $360^\circ$.
5. Square.
6. $120^\circ, 180^\circ, 240^\circ, 300^\circ, 360^\circ$.
7. (i) Yes (ii) No.