Corrigé
9.4 Transformations, Congruence and Similarity
Pack A — Réponses
| # | Question | Réponse |
|---|---|---|
| 1 | Reflect the point $(3, 4)$ in the $x$-axis. | $(3, -4)$ |
| 2 | Reflect the point $(3, 4)$ in the $y$-axis. | $(-3, 4)$ |
| 3 | Translate $(2, -1)$ by the vector $(4, 3)$. State the image. | $(6, 2)$ |
| 4 | Rotate $(3, 0)$ by 90° anticlockwise about the origin. State the image. | $(0, 3)$ |
| 5 | Triangle ABC has $A(1,1), B(2,1), C(1,3)$. Enlarge by sf 2 from the origin. State $A', B', C'$. | $A'(2,2), B'(4,2), C'(2,6)$ |
| 6 | Two figures are similar with scale factor 3. The smaller has length 5 cm. Find the corresponding length on the larger. | 15 cm |
| 7 | Two triangles A and B are similar. A: sides 3, 4, 5. B: shortest side 6. Find B's other sides. | 8 and 10 |
| 8 | Two triangles are congruent. State the four uniqueness rules. | SSS, SAS, ASA, RHS |
| 9 | Reflect the point $(3, 4)$ in the line $y = x$. State the image. | $(4, 3)$ |
| 10 | Two similar shapes have linear scale factor 2. The smaller has area 16 cm². Find the larger area. | 64 cm² |
| 11 | Two similar rectangles have areas 50 cm² and 200 cm². The smaller's shorter side is 5 cm. Find the larger's shorter side. | 10 cm |
| 12 | A triangle has vertices $A(0,0), B(2,0), C(0,2)$. Reflect in the $y$-axis to find $A', B', C'$. | $A'(0,0), B'(-2,0), C'(0,2)$ |
| 13 | Translate $\triangle ABC$ with $A(1, 2), B(3, 2), C(2, 4)$ by vector $(2, -1)$. State the image vertices. | $A'(3,1), B'(5,1), C'(4,3)$ |
| 14 | A square with vertices $A(1,1), B(3,1), C(3,3), D(1,3)$ is enlarged by factor 2 about the origin. State the image vertices. | $A'(2,2), B'(6,2), C'(6,6), D'(2,6)$ |
| 15 | Rotate $\triangle ABC$ with $A(1,1), B(2,1), C(1,3)$ by 90° anticlockwise about the origin. | $A'(-1,1), B'(-1,2), C'(-3,1)$ |
| 16 | Two similar triangles have a side ratio of $\dfrac{x+2}{15} = \dfrac{x}{12}$. Find $x$. | $x = 8$ |
| 17 | Two similar prisms have linear scale factor 3. The smaller has volume 27 cm³. Find the larger volume. | 729 cm³ |
| 18 | A triangle with vertices $(0,0), (4,0), (4,3)$ is reflected in the $x$-axis. State the image area and verify it equals the original. | Both have area 6 — congruent |
| 19 | Two triangles are congruent. State which rule(s) apply when given (a) 3 sides match, (b) 2 sides and the included angle match. | (a) SSS (b) SAS |
| 20 | A square with area 25 cm² is enlarged by scale factor 2. Find the new area and verify the rule. | New area 100 (= 25 × 4) |
| 21 | A triangle ABC: $A(1,1), B(3,1), C(2,3)$. Enlarge by sf 3 from origin. State image vertices. | $A'(3,3), B'(9,3), C'(6,9)$ |
| 22 | Two triangles share sides $a_1, b_1, c_1$ and $a_2, b_2, c_2$ with corresponding sides equal: $a_1 = a_2, b_1 = b_2$, included angle equal. State the congruence rule. | SAS |
| 23 | Triangle ABC is similar to DEF with sf $k = 5/3$ from ABC to DEF. AB = 6, AC = 9. Find DE and DF. | DE = 10, DF = 15 |
| 24 | Two similar shapes have areas 18 cm² and 50 cm². Find the linear scale factor. | $\sqrt{50/18} = 5/3$ |
| 25 | A triangle has sides 6 cm, 8 cm, 10 cm. A similar triangle has perimeter 36 cm. Find its sides. | 9, 12, 15 |
| 26 | Triangle ABC has $A(0, 0), B(4, 0), C(0, 3)$. Rotate 90° clockwise about origin. State the image vertices. | $A'(0,0), B'(0,-4), C'(3,0)$ |
| 27 | Two similar prisms have surface areas $24$ cm² and $96$ cm². Find the volume ratio. | 1 : 8 |
| 28 | A rectangle 4 × 6 is enlarged by sf 1.5. Find (a) new dimensions, (b) new area, (c) new perimeter. | (a) 6 × 9 (b) 54 (c) 30 |
| 29 | A triangle has vertices $A(1, 2), B(4, 2), C(4, 6)$. Reflect in $y = x$ to get image $A', B', C'$. | $A'(2,1), B'(2,4), C'(6,4)$ |
| 30 | A triangle PQR has $PQ = 6, QR = 8, RP = 10$. Show ΔPQR is right-angled. Identify the right angle. | Yes; right angle at Q |
| 31 | A cylindrical glass of radius 4 cm and height 10 cm is full of water. Water is poured into a similar cylindrical glass (radius half of the first). To what height does the water rise? | 40 cm |
| 32 | Triangle ABC is similar to triangle PQR with sf 2 from ABC to PQR. Triangle ABC has area 18 cm². Find PQR's area and perimeter ratio. | PQR area 72; perim ratio 1:2 |
| 33 | A photo is enlarged so its area is doubled. Find the exact linear scale factor. | $\sqrt{2}$ |
| 34 | A triangle is rotated 90° about the origin, then reflected in the $x$-axis. The original is $A(1, 2)$. Find the final image. | $(-2, -1)$ |
| 35 | Two similar cones have heights 6 and 9. The smaller has slant 8. Find the larger's slant. | 12 |
| 36 | Triangle ABC has angles 60°, 80°, 40°. Triangle DEF has angles 60°, 80°, 40°. Are they similar? Are they congruent? | Similar (AAA); not necessarily congruent |
| 37 | A triangle has area 24 cm². It is enlarged by sf 2.5 from a point. Find the area of the image. | 150 cm² |
| 38 | Triangle ABC has $A(0,0), B(3,0), C(0,4)$. It is enlarged by sf 2 from the centre $(1, 1)$. Find the image vertices. | $A'(-1, -1), B'(5, -1), C'(-1, 7)$ |
| 39 | Two similar triangles have linear scale factor 4:7. Their corresponding areas are in what ratio? | 16 : 49 |
| 40 | A figure of area $A$ is transformed by rotation. Show that the area is preserved. | Rotation is an isometry — preserves all distances, hence all areas |
Pack B — Réponses
| # | Question | Réponse |
|---|---|---|
| 1 | Reflect the point $(-2, 5)$ in the $x$-axis. | $(-2, -5)$ |
| 2 | Reflect the point $(-2, 5)$ in the $y$-axis. | $(2, 5)$ |
| 3 | Translate $(2, -1)$ by the vector $(4, 3)$. State the image. | $(3, 3)$ |
| 4 | Rotate $(3, 0)$ by 90° anticlockwise about the origin. State the image. | $(-4, 0)$ |
| 5 | Triangle ABC has $A(1,1), B(2,1), C(1,3)$. Enlarge by sf 2 from the origin. State $A', B', C'$. | $A'(3,3), B'(6,3), C'(3,9)$ |
| 6 | Two figures are similar with scale factor 3. The smaller has length 5 cm. Find the corresponding length on the larger. | 24 cm |
| 7 | Two triangles A and B are similar. A: sides 3, 4, 5. B: shortest side 6. Find B's other sides. | 24 and 26 |
| 8 | Two triangles are congruent. State the four uniqueness rules. | Same. |
| 9 | Reflect the point $(3, 4)$ in the line $y = x$. State the image. | $(-2, 5)$ |
| 10 | Two similar shapes have linear scale factor 2. The smaller has area 16 cm². Find the larger area. | 81 cm² |
| 11 | Two similar rectangles have areas 18 cm² and 162 cm². The smaller's shorter side is 3 cm. Find the larger's shorter side. | 9 cm |
| 12 | A triangle has vertices $A(0,0), B(2,0), C(0,2)$. Reflect in the $y$-axis to find $A', B', C'$. | $A'(0,0), B'(0,2), C'(2,0)$ |
| 13 | Translate $\triangle ABC$ with $A(1, 2), B(3, 2), C(2, 4)$ by vector $(2, -1)$. State the image vertices. | $A'(-2,4), B'(0,4), C'(-1,6)$ |
| 14 | A square with vertices $A(1,1), B(3,1), C(3,3), D(1,3)$ is enlarged by factor 2 about the origin. State the image vertices. | $A'(0.5,0.5), B'(1.5,0.5), C'(1.5,1.5), D'(0.5,1.5)$ |
| 15 | Rotate $\triangle ABC$ with $A(1,1), B(2,1), C(1,3)$ by 90° anticlockwise about the origin. | $A'(-1,-1), B'(-2,-1), C'(-1,-3)$ |
| 16 | Two similar triangles have a side ratio of $\dfrac{x+2}{15} = \dfrac{x}{12}$. Find $x$. | $x = 3$ |
| 17 | Two similar prisms have linear scale factor 3. The smaller has volume 27 cm³. Find the larger volume. | 400 cm³ |
| 18 | A triangle with vertices $(0,0), (4,0), (4,3)$ is reflected in the $x$-axis. State the image area and verify it equals the original. | Same. |
| 19 | Two triangles are congruent. State which rule(s) apply when given (a) 3 sides match, (b) 2 sides and the included angle match. | Same. |
| 20 | A square with area 25 cm² is enlarged by scale factor 2. Find the new area and verify the rule. | 324 (= 36 × 9) |
| 21 | A triangle ABC: $A(1,1), B(3,1), C(2,3)$. Enlarge by sf 2 from origin. State image vertices. | $A'(2,2), B'(6,2), C'(4,6)$ |
| 22 | Two triangles share sides $a_1, b_1, c_1$ and $a_2, b_2, c_2$ with corresponding sides equal: $a_1 = a_2, b_1 = b_2$, included angle equal. State the congruence rule. | Same. |
| 23 | Triangle ABC is similar to DEF with sf $k = 5/3$ from ABC to DEF. AB = 6, AC = 9. Find DE and DF. | DE = 14, DF = 21 |
| 24 | Two similar shapes have areas 18 cm² and 50 cm². Find the linear scale factor. | $\sqrt{75/12} = 5/2$ |
| 25 | A triangle has sides 6 cm, 8 cm, 10 cm. A similar triangle has perimeter 36 cm. Find its sides. | 10, 24, 26 |
| 26 | Triangle ABC has $A(0, 0), B(4, 0), C(0, 3)$. Rotate 90° clockwise about origin. State the image vertices. | $A'(0,0), B'(0,4), C'(-3,0)$ |
| 27 | Two similar prisms have surface areas $24$ cm² and $96$ cm². Find the volume ratio. | $\sqrt{3^3/5^3}$... see working |
| 28 | A rectangle 4 × 6 is enlarged by sf 1.5. Find (a) new dimensions, (b) new area, (c) new perimeter. | (a) 10 × 16 (b) 160 (c) 52 |
| 29 | A triangle has vertices $A(1, 2), B(4, 2), C(4, 6)$. Reflect in $y = x$ to get image $A', B', C'$. | Same. |
| 30 | A triangle PQR has $PQ = 6, QR = 8, RP = 10$. Show ΔPQR is right-angled. Identify the right angle. | Yes; right angle at Q |
| 31 | A cylindrical glass of radius 6 cm and height 12 cm is full of water. Water is poured into a similar cylindrical glass (radius half of the first). To what height does the water rise? | 48 cm |
| 32 | Triangle ABC is similar to triangle PQR with sf 2 from ABC to PQR. Triangle ABC has area 18 cm². Find PQR's area and perimeter ratio. | PQR 108; ratio 1:3 |
| 33 | A photo is enlarged so its area is doubled. Find the exact linear scale factor. | $\sqrt{3}$ |
| 34 | A triangle is rotated 90° about the origin, then reflected in the $x$-axis. The original is $A(1, 2)$. Find the final image. | $(1, -2)$ |
| 35 | Two similar cones have heights 6 and 9. The smaller has slant 8. Find the larger's slant. | 16.8 |
| 36 | Triangle ABC has angles 60°, 80°, 40°. Triangle DEF has angles 60°, 80°, 40°. Are they similar? Are they congruent? | Same. |
| 37 | A triangle has area 24 cm². It is enlarged by sf 2.5 from a point. Find the area of the image. | 450 cm² |
| 38 | Triangle ABC has $A(0,0), B(3,0), C(0,4)$. It is enlarged by sf 2 from the centre $(1, 1)$. Find the image vertices. | $A'(0,0), B'(9,0), C'(0,12)$ |
| 39 | Two similar triangles have linear scale factor 4:7. Their corresponding areas are in what ratio? | 25 : 64 |
| 40 | A figure of area $A$ is transformed by rotation. Show that the area is preserved. | Same for reflection. |
Problèmes — Solutions détaillées
**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.
(a) 6 (b) $A'(1,-1), B'(4,-1), C'(1,-5)$ (c) $A'(-1,1), B'(-1,4), C'(-5,1)$ (d) $A'(3,-2), B'(6,-2), C'(3,2)$
**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 $b$.
(a) 36 cm² (b) 40 cm (c) $(2a/3) \times (2b/3)$
**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.
(a) 8 (b) 200 cm³ (c) 120 cm²
**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?
(a) SAS (b) $AC = PR$ (equal) (c) Yes — congruent ⇒ similar (sf 1)
**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.
Step 1: $(-2, 3)$. Step 2: $(-3, -2)$. Step 3: $(-2, -4)$.
**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.
(a) Both have a vertical line, horizontal shadow, same sun angle (AAA) (b) 21 m (c) ≈ 71.6°
**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.
(a) $64:343$ (b) 343 cm³ (c) 80 cm²
**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.
(a) $A'(2,-1), B'(5,-1), C'(2,-4)$ (b) $A''(-2,1), B''(-5,1), C''(-2,4)$ (c) Reflection in the $y$-axis
**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.
(a) 2 × 10⁶ m² (b) 2 km²
**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.
(a) Both fully determined (b) Only if B's third side = 7 (c) Use cosine rule to find B's third side: $c^2 = 25 + 36 - 60\cos 40° \approx 61 - 45.96 \approx 15.04$, $c \approx 3.88$. Not 7.
**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.
(a) $A'(4, 1), B'(13, 1), C'(7, 10)$ (b) 40.5
**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.
(a) 6 (b) 6 (c) Triangle: order 3, 3 lines