Area of Rhombuses, Kites and Trapeziums — Solutions

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1. Area of rhombuses Fluency

   1. 30 cm²
   2. 56 m²
   3. 40.5 cm²
   4. 150 mm²
   5. 11.2 m²
   6. 99 cm²
   7. 22.2 m²
   8. 330 cm²

2. Area of kites Fluency

   1. 42 cm²
   2. 72 m²
   3. 20 cm²
   4. 120 mm²
   5. 44 m²
   6. 16 cm²
   7. 4.5 m²
   8. 90 cm²

3. Area of trapeziums Fluency

   1. 30 cm²
   2. 63 m²
   3. 40 cm² — this is a rectangle (both “parallel sides” are equal)
   4. 96 mm²
   5. 36 m²
   6. 153 cm²
   7. 38.4 m²
   8. 97.5 cm²

4. Mixed — identify and apply Fluency

   1. Rhombus; A = ½ × 16 × 10 = 80 cm²
   2. Trapezium; A = ½ × (9+5) × 4 = 28 m²
   3. Kite; A = ½ × 14 × 8 = 56 cm²
   4. Trapezium; A = ½ × (11+7) × 6 = 54 mm²
   5. Rhombus (square); A = ½ × 9 × 9 = 40.5 m²
   6. Trapezium; A = ½ × (20+8) × 7 = 98 cm²
   7. Kite; A = ½ × 18 × 13 = 117 m²
   8. Trapezium; A = ½ × (6.5+3.5) × 4 = 20 cm²

5. Find missing measurements Understanding

   1. d₂ = 8 cm  (48 × 2 ÷ 12)
   2. d₂ = 7 m  (35 × 2 ÷ 10)
   3. h = 6 cm  (54 × 2 ÷ (10+8))
   4. Second parallel side = 13 m  (66 × 2 ÷ 6 − 9 = 22 − 9)
   5. Each diagonal = √180 ≈ 13.4 mm  (½ × d × d = 90; d² = 180)
   6. d₁ = 2d₂; ½ × 2d₂ × d₂ = 120; d₂² = 120; d₂ ≈ 11.0 cm, d₁ ≈ 21.9 cm
   7. Sides are 3k and 5k; ½(3k+5k)×8=80; 4k=10; k=2.5; sides = 7.5 m and 12.5 m
   8. Parallelogram: A = base × h; 96 = 12 × h; h = 8 cm. One trapezium = 48 cm²

6. Problem Solving Problem Solving

   1. A = ½ × 60 × 45 = 1350 cm² = 0.135 m²
   2. Edging cost = 26 × $12 = $312
   3. 1. A = ½ × (48+36) × 25 = 1050 m²
      2. 0.105 ha
      3. $892 500  (1050 × $850)
   4. Rectangle: 10 × 6 = 60 cm². Trapezium: ½×(10+6)×4 = 32 cm². Total = 92 cm²

7. Area from diagrams Understanding

   1. 104 cm²  (½(10+16)×8)
   2. 63 cm²  (½×14×9)
   3. 108 cm²  (½×12×18)

8. Formula table Understanding

   1. 20 cm²
   2. 35 m²
   3. 28 cm²
   4. 72 mm²
   5. 216 m²

9. Spot the error Understanding

   1. Taylor forgot to halve. Correct: A = ½(8+6)×5 = 35 cm²
   2. Lee forgot to halve. Correct: A = ½×10×4 = 20 m²
   3. Morgan added diagonals instead of multiplying. Correct: A = ½×6×4 = 12 cm²
   4. Sam used slant height (6 m) not perpendicular height (5 m). Correct: A = ½(12+8)×5 = 50 m²

10. Extended response Problem Solving

    1. 1. 48 cm²  (½×12×8)
       2. 75 tiles  (3600 ÷ 48)
       3. $108.75  (75 × $1.45)
    2. 1. 12 m²  (½×6×4)
       2. 15 m²  (½(6+4)×3)
       3. $216  (27 m² × $8)