Enlargement and Scale Factor — Solutions

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1. Calculate the scale factor

   1. 5 → 15: k = 3
   2. 8 → 4: k = 0.5
   3. 6 → 9: k = 1.5
   4. 12 → 3: k = 0.25
   5. 7 → 21: k = 3
   6. 20 → 5: k = 0.25
   7. 9 → 6: k = ⅔ ≈ 0.667
   8. 4 → 10: k = 2.5

2. Find unknown side lengths

   1. k=3, original 7: 7 × 3 = 21 cm
   2. k=0.5, original 14: 14 × 0.5 = 7 cm
   3. k=15/5=3, original 8: 8 × 3 = 24 cm
   4. k=12/4=3, original 9: 9 × 3 = 27 cm
   5. k=9/6=1.5, sides 8 and 10: 8 × 1.5 = 12 cm; 10 × 1.5 = 15 cm
   6. k=0.5, sides 14, 20, 24: 7 cm, 10 cm, 12 cm

3. Similar, congruent or neither?

   1. 3,4,5 and 6,8,10: Similar (scale factor 2; all sides in ratio 1:2)
   2. 5×8 and 10×16: Similar (scale factor 2)
   3. 3×6 and 4×8: Similar (both have ratio 1:2, scale factor 4/3)
   4. Squares 5 cm and 7 cm: Similar (all squares are similar); not congruent
   5. 4,7,9 and 8,12,16: Neither (ratios: 8/4=2, 12/7≈1.71, 16/9≈1.78 — not equal)
   6. Equilateral 3 cm and 9 cm: Similar (scale factor 3; all equilateral triangles are similar)
   7. Same angles, different sides: Similar (same angles means AA similarity; sides in ratio 5:3)

4. Find unknowns in similar figures

   1. Scale factor 2, sides 5,12,x and 10,24,26: k = 10/5 = 2. x = 26 ÷ 2 = 13 cm
   2. k=20/8=2.5, side 6: 6 × 2.5 = 15 cm
   3. k=25/10=2.5, length 15: 15 × 2.5 = 37.5 cm
   4. Scale 1:24, car 4.8 m: 4.8 m = 480 cm. Model = 480 ÷ 24 = 20 cm
   5. Perimeters 30 and 45, k=45/30=1.5, side 8: k = 1.5. 8 × 1.5 = 12 cm

5. Problem solving

   1. Scale drawing:
      1. 8.5 m ÷ 2.5 = 3.4 cm on the drawing
      2. 4.2 cm × 2.5 = 10.5 m; 3.6 cm × 2.5 = 9 m. Room is 10.5 m × 9 m
   2. Tree shadow: Scale factor = 9/2.25 = 4 (shadow ratio). Tree height = 1.5 × 4 = 6 m
   3. Photo enlargement:
      1. k = 20 ÷ 6 = 3.33 (max scale factor)
      2. Length = 9 × 3.33 = 30 cm
      3. No, the paper would need to be at least 30 cm long. A 20 cm paper is not long enough.
   4. Map scale 1:50 000:
      1. 8.4 cm × 50 000 = 420 000 cm = 4.2 km
      2. 12.5 cm × 50 000 = 625 000 cm = 6.25 km

6. Effect of scale factor on perimeter and area

   1. Rectangle: perimeter 24 cm, area 32 cm², k = 3: New perimeter = 24 × 3 = 72 cm. New area = 32 × 3² = 32 × 9 = 288 cm²
   2. Area 20 → 180, find k: k² = 180 ÷ 20 = 9; k = 3
   3. Square side 5 cm, k = 2.5: New side = 5 × 2.5 = 12.5 cm. New perimeter = 4 × 12.5 = 50 cm. New area = 5² × 2.5² = 25 × 6.25 = 156.25 cm²
   4. Area 48 m², k = 0.5: New area = 48 × 0.5² = 48 × 0.25 = 12 m²
   5. Areas 25 and 100 cm²: k² = 100 ÷ 25 = 4; k = 2

7. Enlargements on the coordinate plane

   1. Triangle (1,1),(3,1),(1,4) × k=2 from origin: (2, 2), (6, 2), (2, 8)
   2. Rectangle (0,0),(4,0),(4,2),(0,2), k=1.5: (0, 0), (6, 0), (6, 3), (0, 3)
   3. Image P=(6,9), k=3 from origin: Q = (6 ÷ 3, 9 ÷ 3) = (2, 3)
   4. Triangle (2,2),(6,2),(2,6), k=0.5: Image: (1,1), (3,1), (1,3). This is a reduction (k < 1).
   5. Original (3,4), image (9,12): k = 9 ÷ 3 = 3 (check: 12 ÷ 4 = 3 ✓)

8. Multi-step scale factor problems

   1. Scale 1:72, wingspan 36 m: 36 m = 3600 cm. Model = 3600 ÷ 72 = 50 cm
   2. Floor plan 1:200, room 3.5 × 4 cm:
      1. 3.5 × 200 = 700 cm = 7 m; 4 × 200 = 800 cm = 8 m
      2. Real area = 7 × 8 = 56 m²
      3. Plan area = 3.5 × 4 = 14 cm². Real area is 200² = 40 000 times larger than plan area (56 m² = 560 000 cm²; 560 000 ÷ 14 = 40 000).
   3. Similar triangles, sides 8 and 20 cm:
      1. k = 20 ÷ 8 = 2.5
      2. Perimeter = 28 × 2.5 = 70 cm
      3. Area = 24 × 2.5² = 24 × 6.25 = 150 cm²

9. Reasoning about similarity and scale

   1. Flagpole 6 m shadow, post 1.8 m → 0.9 m shadow: k = 6 ÷ 0.9 = 6.&overline;6. Height = 1.8 × (6 ÷ 0.9) = 1.8 × 20/3 = 12 m. (Using ratio: height/shadow equal: h/6 = 1.8/0.9 = 2, so h = 12 m)
   2. Photo 8 × 12 cm enlarged to length 30 cm: k = 30 ÷ 12 = 2.5. New width = 8 × 2.5 = 20 cm. A 20 cm wide paper is exactly wide enough (just fits).
   3. Pentagons perimeters 45 and 75 cm:
      1. k = 75 ÷ 45 = 5/3 ≈ 1.67
      2. 9 × 5/3 = 15 cm
      3. Area = 81 × (5/3)² = 81 × 25/9 = 225 cm²
   4. Painting 60 × 45 cm reduced to 10 cm wide:
      1. k = 10 ÷ 60 = 1/6
      2. Height = 45 × 1/6 = 7.5 cm
      3. Area fraction = (1/6)² = 1/36 of the original area

10. Successive enlargements

    1. k=2 then k=3: Combined scale factor = 2 × 3 = 6
    2. 5 cm, k=2 then k=1.5:
       1. 5 × 2 = 10 cm
       2. 10 × 1.5 = 15 cm
       3. 5 × (2 × 1.5) = 5 × 3 = 15 cm ✓
    3. Square 4 cm, k=2.5 then k=0.4:
       1. 4 × 2.5 × 0.4 = 4 cm
       2. Combined k = 2.5 × 0.4 = 1
       3. The overall scale factor is 1, so the shape returns to its original size (congruent to the original).
    4. Photo 6 × 9 cm, enlarged to 21 cm wide, then k=½: k₁ = 21 ÷ 6 = 3.5. Enlarged: 21 × 31.5 cm. Then × ½: final = 10.5 cm × 15.75 cm