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Problem-solving

9.4 Transformations, Congruence and Similarity

Show all working. Partial marks are given for method.

  1. 1
    **Triangle on coordinate plane.** Triangle ABC has vertices $A(1, 1)$, $B(4, 1)$, $C(1, 5)$. (a) Find the area. (b) Reflect in the $x$-axis. State the image vertices. (c) Rotate 90° anticlockwise about the origin. State the image vertices. (d) Translate by vector $(2, -3)$. State the image vertices.

    Working space

  2. 2
    **Similar figures.** Two similar rectangles have linear scale factor 3:2. (a) The larger has area 81 cm². Find the smaller area. (b) The larger has perimeter 60 cm. Find the smaller perimeter. (c) If the larger has dimensions $a$ × $b$, express the smaller in terms of $a$ and $b$.

    Working space

  3. 3
    **Cone scaling.** Two similar cones have heights 5 cm and 10 cm. (a) Find the volume scale factor. (b) The smaller has volume 25 cm³. Find the larger volume. (c) The smaller has surface area 30 cm². Find the larger SA.

    Working space

  4. 4
    **Triangle congruence.** Triangles ABC and PQR have $AB = PQ = 6$, $BC = QR = 8$, and the included angle $B = Q = 50°$. (a) State the congruence rule. (b) State the relationship between the third sides. (c) Are these triangles similar?

    Working space

  5. 5
    **Transformation composition.** A point $P(2, 3)$ is reflected in the $y$-axis, then rotated 90° anticlockwise about the origin, then translated by $(1, -2)$. (a) Find the position after each step. (b) Find the final image.

    Working space

  6. 6
    **Similar triangles in a real-world scenario.** A tree of height 12 m casts a shadow 4 m long, while a flagpole of unknown height casts a shadow 7 m long. (a) Why are the shadow triangles similar? (b) Find the flagpole's height. (c) Find the angle of elevation of the sun.

    Working space

  7. 7
    **Volume scaling.** Two similar pyramids have linear scale factor 4:7. (a) Find the volume scale factor. (b) The smaller pyramid has volume 64 cm³. Find the larger. (c) The larger pyramid has surface area 245 cm². Find the smaller SA.

    Working space

  8. 8
    **Two-step transformation challenge.** A triangle has vertices $A(2, 1), B(5, 1), C(2, 4)$. (a) Reflect in the $x$-axis: state image vertices. (b) Then rotate 180° about the origin. (c) Identify the single transformation that combines both into one.

    Working space

  9. 9
    **Map scale and similarity.** A 1:50 000 map shows a park of area 8 cm². Find the real area in (a) m², (b) km².

    Working space

  10. 10
    **Congruence test puzzle.** Two triangles have: A: sides 5, 6, 7 B: sides 5, 6, included angle of 40° (a) Which uniqueness rules might apply? (b) Are they necessarily congruent? (c) Compute the third side of triangle B using the cosine rule (or Pythagoras-like reasoning), and decide whether it could equal 7.

    Working space

  11. 11
    **Enlargement from a centre.** Triangle ABC has $A(2, 1), B(5, 1), C(3, 4)$. Enlarge by sf 3 from centre $P(1, 1)$. (a) Find the image vertices. (b) Find the image area, given the original area is 4.5.

    Working space

  12. 12
    **Symmetry investigation.** A regular hexagon has rotational and reflective symmetries. (a) State the order of rotational symmetry. (b) State the number of lines of reflective symmetry. (c) Compare with an equilateral triangle.

    Working space