Problems — 9.4 Transformations, Congruence and Similarity

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.

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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 $b$.

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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.

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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?

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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