Topic Review — Congruence and Transformations — Solutions

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1. Congruent figures and congruence tests

   1. Four tests: SSS (three sides), SAS (two sides and included angle), AAS (two angles and corresponding side), RHS (right angle, hypotenuse, side)
   2. Sides 9,12,15: SSS. ▵ABC ≅ ▵DEF (or whichever matching order is given)
   3. ▵PQR ≅ ▵XYZ: XY = PQ = 7 cm; ∠X = ∠P = 48°
   4. Two equal angles: No. Two triangles with two equal angles are similar but not necessarily congruent (their sizes may differ).
   5. RS=UV, ST=VW, ∠RST=∠UVW: SAS. ▵RST ≅ ▵UVW (SAS)

2. Transformations — finding images

   1. A(2,−3) + (−5,4): A′(−3, 1)
   2. B(4,6) reflect x-axis: B′(4, −6)
   3. C(−3,1) reflect y-axis: C′(3, 1)
   4. D(5,2) 90° anticlockwise: D′(−2, 5)
   5. E(3,−4) 180°: E′(−3, 4)
   6. F(7,3) reflect in y=x: F′(3, 7)

3. Scale factor and similar figures

   1. k=10/4=2.5, sides: k = 2.5. Other sides: 6×2.5=15 cm, 8×2.5=20 cm
   2. k=2.5, rectangle: 5×2.5=12.5 cm wide; 12×2.5=30 cm long
   3. 1:200, ship 160 m: 160 m = 16 000 cm. Model = 16 000 ÷ 200 = 80 cm
   4. Area ratio 9:25: Scale factor = √(9/25) = 3/5 = 0.6

4. Describe and apply transformations

   1. A to B, y-coordinates negated: Reflection in the x-axis
   2. C(2,1),(5,1),(5,4) to D(−1,2),(−1,5),(−4,5): Rotation of 90° anticlockwise about the origin
   3. P(4,3) translate then reflect y-axis: After translation: (−2, 5). After reflect y-axis: P′′(2, 5)
   4. Mirror image: Reflection. Real-life example: a butterfly’s wings, a face reflected in water, a logo reflected to create a symmetric design.

5. Problem solving

   1. Parallelogram ABCD:
      1. AB = CD (opposite sides), BC = DA (opposite sides), AC = AC (common/shared). ∠BAC = ∠DCA (alternate angles, AB∥CD). ∠BCA = ∠DAC (alternate angles, BC∥DA).
      2. AB = CD, BC = DA, AC = CA (common). ∴ ▵ABC ≅ ▵CDA (SSS)
   2. Photo enlargement:
      1. k = 30 ÷ 8 = 3.75
      2. Length = 12 × 3.75 = 45 cm
      3. Yes, the enlarged photo is similar to the original. Both dimensions are multiplied by the same scale factor (3.75), so the ratio of sides is preserved.
   3. P(3,4) reflect in y=x then rotate 90° CW: After reflection in y=x: (4,3). After 90° CW rotation (x,y)→(y,−x): (3,−4). If we reverse the order (rotate first, then reflect): rotate (3,4) 90° CW → (4,−3); reflect in y=x: (−3,4). The results differ, so order matters.