Properties of Quadrilaterals — Solutions

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1. Name the quadrilateral

   1. All sides equal, all angles 90°: Square
   2. Opposite sides parallel and equal, all angles 90°: Rectangle
   3. All sides equal, opposite angles equal, no right angles: Rhombus
   4. Exactly one pair of parallel sides: Trapezium
   5. Two pairs of adjacent equal sides, one axis of symmetry: Kite
   6. Opposite sides parallel and equal, opposite angles equal: Parallelogram
   7. Diagonals equal, bisect at right angles, all sides equal: Square
   8. One pair parallel sides, two pairs equal angles: Isosceles trapezium

2. State yes or no with reason

   1. Rectangle, all sides equal?: No — a rectangle only has opposite sides equal; all four sides equal only in a square.
   2. Rhombus, all angles 90°?: No — a rhombus has opposite equal angles that are not necessarily 90°; only a square has all 90° in that family.
   3. Parallelogram, diagonals bisect each other?: Yes — this is a defining property of parallelograms.
   4. Square, diagonals perpendicular?: Yes — the diagonals of a square bisect each other at right angles.
   5. Kite, both pairs opposite sides parallel?: No — a kite has two pairs of adjacent equal sides, not parallel opposite sides.
   6. Trapezium, exactly one pair parallel?: Yes — by definition, a trapezium has exactly one pair of parallel sides.
   7. Rectangle, diagonals equal?: Yes — the diagonals of a rectangle are equal in length.
   8. Rhombus, diagonals equal?: No — the diagonals of a rhombus are only equal if it is also a square.

3. Find unknown angles

   1. Parallelogram, one angle 75°: Opposite angle = 75°. Co-interior angles add to 180°: 180−75=105°. Angles: 75°, 105°, 75°, 105°
   2. Rhombus, one angle 40°: Opposite = 40°. Adjacent = 180−40 = 140°. Angles: 40°, 140°, 40°, 140°
   3. Trapezium, 65°+115°+65°+?: 360−65−115−65 = 115°
   4. Rectangle diagonal 38° to base: Angles at centre: 38°, 52° (90−38), 38°, 52° — the diagonals of a rectangle bisect each other (not at right angles unless it’s a square)
   5. Kite angles 110°, 70°, and two unknowns: 360 − 110 − 70 = 180° for both equal angles. Each = 90°

4. Diagonal properties and lengths

   1. Rectangle diagonal 13 cm: Half-diagonal = 13 ÷ 2 = 6.5 cm
   2. Parallelogram diagonals 18 and 24: Half of 18 = 9 cm; half of 24 = 12 cm from centre to each vertex
   3. Rhombus diagonals 10 and 24: Half-diagonals: 5 and 12. Side = √(5² + 12²) = √(25+144) = √169 = 13 cm
   4. Square diagonal 10√2: Side = diagonal ÷ √2 = 10√2 ÷ √2 = 10 cm
   5. Kite non-axis diagonal 10 cm: Bisected into two equal halves: each section = 5 cm

5. Classify the quadrilateral

   1. All sides equal, all angles 90°, opposite sides parallel: Square
   2. Both pairs opposite sides parallel, ∠A = 72°: Parallelogram
   3. AB ∥ CD only, co-interior angles: Trapezium
   4. Two pairs adjacent equal sides, perpendicular diagonals: Kite
   5. All 90°, opposite sides equal but unequal pairs: Rectangle

6. Problem solving

   1. Rhombus park, diagonals 40 m and 30 m: Total path = 40 + 30 = 70 m. Area = (40 × 30) ÷ 2 = 600 m²
   2. Parallelogram, ∠A = 3x+15, ∠C = 5x−11: Opposite angles equal: 3x+15 = 5x−11, 26 = 2x, x = 13. ∠A = ∠C = 54°. ∠B = ∠D = 180−54 = 126°
   3. Rectangle perimeter 38: 2(2y+5) + 2(y+1) = 38, 4y+10+2y+2 = 38, 6y = 26, y = 13/3 ≈ 4.33. Length = 2(4.33)+5 ≈ 13.67 m; Width = 4.33+1 ≈ 5.33 m
   4. Kite area:
      1. Area = (10 × 6) ÷ 2 = 30 cm²
      2. Split into two triangles along the axis: each has base 6 cm and combined height 10 cm. Upper triangle: ½×6×(upper part of axis). Lower: ½×6×(lower). Total = ½×6×10 = 30 cm² ✓

7. Algebraic angle problems

   1. Parallelogram (4x+10)° = (6x−20)°: Opposite angles equal: 4x+10 = 6x−20; 30 = 2x; x = 15. Angles: 70°, 110°, 70°, 110°
   2. Rhombus (2y+15)° + (5y−12)° = 180°: Co-interior: 7y + 3 = 180; 7y = 177; y = 177/7 ≈ 25.3°. Angles: (2×177/7+15) ≈ 65.6° and 114.4° × 2 each.
   3. Trapezium (3a+5) + (2a+15) = 180: 5a + 20 = 180; 5a = 160; a = 32. Angles: 3(32)+5 = 101° and 2(32)+15 = 79°
   4. Kite ∠A=90°, ∠C=90°, ∠B=3∠D: 90+90+∠B+∠D = 360; ∠B+∠D = 180; 3∠D+∠D = 180; 4∠D = 180; ∠D = 45°; ∠B = 135°

8. Perimeter and side lengths

   1. Parallelogram perimeter 60: 2(3x+2) + 2(x+10) = 60; 6x+4+2x+20 = 60; 8x = 36; x = 4.5. Sides: 15.5 cm and 14.5 cm
   2. Rhombus sides equal: 2m+5 = 4m−3; 8 = 2m; m = 4. Side = 13 cm. Perimeter = 52 cm
   3. Rectangle perimeter 54: 2(5y−1) + 2(2y+3) = 54; 10y−2+4y+6 = 54; 14y = 50; y = 25/7 ≈ 3.57. Length ≈ 16.86 cm; Width ≈ 10.14 cm
   4. Kite perimeter 72: 2(3n+4) + 2(n+8) = 72; 6n+8+2n+16 = 72; 8n = 48; n = 6. Longer sides = 22 cm; shorter sides = 14 cm

9. Diagonal and geometric reasoning

   1. Square diagonal 20 cm: d = s√2; s = 20 ÷ √2 = 10√2 ≈ 14.14 cm
   2. Rhombus diagonals 16 and 12: Half-diagonals: 8 and 6. Side = √(8²+6²) = √(64+36) = √100 = 10 cm
   3. Rectangle diagonal 26, side 10: Other side = √(26²−10²) = √(676−100) = √576 = 24 cm. Perimeter = 2(10+24) = 68 cm. Area = 10×24 = 240 cm²
   4. Kite diagonals 18 and 10: Area = (18×10)÷2 = 90 cm². Shorter diagonal bisected at 6 and 4 from the axis. Upper sides: √(6²+6²)=6√2≈8.49 cm (2 sides). Lower sides: √(12²+5²)=13 cm; and √(6²+5²)=√61≈7.81 cm — see note: axis splits at a point 6 cm from one end, so halves of short diagonal are 5 each from the axis meeting point. Upper triangle legs: 6 and 5 → √61 ≈ 7.81 cm (2 sides). Lower triangle legs: 12 and 5 → 13 cm (2 sides).

10. Real-world and investigation

    1. Parallelogram 120 m and 85 m: Perimeter = 2(120 + 85) = 2 × 205 = 410 m of fencing
    2. Rhombus diagonals 24 and 10:
       1. Area = (24×10) ÷ 2 = 120 cm²
       2. Half-diagonals 12 and 5. Side = √(12²+5²) = √169 = 13 cm
       3. Perimeter = 4 × 13 = 52 cm
    3. PQRS angles: (2x+5)+(x+20)+(3x−10)+(2x+15) = 360; 8x+30 = 360; 8x = 330; x = 41.25. ∠P = 87.5°; ∠Q = 61.25°; ∠R = 113.75°; ∠S = 97.5°. No pairs of equal opposite angles and no right angles, so it is a general (irregular) quadrilateral.
    4. Square tiles, diagonal 28 cm:
       1. Side = 28 ÷ √2 = 14√2 ≈ 19.8 cm
       2. Floor = 420 cm × 360 cm. Tiles per row = 420 ÷ 19.8 ≈ 21.2 → 22; tiles per column = 360 ÷ 19.8 ≈ 18.2 → 19. Total = 22 × 19 = 418 tiles