Finding Unknown Sides in Similar Figures — Solutions

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1. Find the value of x

   1. k = 6/4 = 1.5. x = 10 × 1.5 = 15 cm
   2. k = 12/8 = 1.5. x = 5 × 1.5 = 7.5 m
   3. k = 10.5/7 = 1.5. TU = 9 × 1.5 = 13.5 cm
   4. k = 4/6 = 2/3 (reduction). x = 9 × (2/3) = 6 cm
   5. k = 15/10 = 1.5. Corresponding larger side = 8 × 1.5 = 12

2. ▵ABC ~ ▵DEF (AB=6, BC=9, CA=12, DE=4)

   1. k = DE/AB = 4/6 = 2/3
   2. EF = BC × k = 9 × (2/3) = 6
   3. FD = CA × k = 12 × (2/3) = 8
   4. Perimeter ▵ABC = 6+9+12 = 27. Perimeter ▵DEF = 4+6+8 = 18.
   5. Ratio = 18/27 = 2/3 = k. The perimeter scale factor equals k. ✓

3. Area scale factor

   1. Area scale factor = k² = 9. Larger area = 8 × 9 = 72 cm²
   2. k² = 200/50 = 4. k = 2
   3. k² = 6.25. Smaller area = 125/6.25 = 20 m²
   4. k = 4. k² = 16. Enlarged area = 24 × 16 = 384 cm²

4. Set up and solve proportions

   1. k = PQ/LM = QR/MN: 8/x = 16/12. Cross multiply: 16x = 96. x = 6 cm
   2. k = EF/AB = 6/10 = 0.6. x = BC × k = x is in ABCD; BC/FG = AB/EF → BC = x, FG = 9. x/9 = 10/6 → x = 15. y = GH × (10/6) — wait: EFGH ~ ABCD so EF/AB = 6/10. y = GH corresponds to CD: y/CD — actually use k = 6/10 = 3/5. BC corresponds to FG: FG = BC×(3/5), 9 = x×(3/5), x = 15. GH corresponds to CD: y = 15×(3/5) = 9.
   3. Triangles are similar (AB ∥ XY not given, but ▵XYZ ~ ▵XAB). XA/XY = XB/XZ. 4/10 = 6/XZ. XZ = 60/4 = 15.
   4. k = 15/10 = 1.5. Base of second = 8 × 1.5 = 12 cm

5. Similar trapezoids

   1. k = 9/6 = 1.5 (check: 15/10 = 1.5 ✓). k = 1.5
   2. Non-parallel sides = 5 × 1.5 = 7.5 cm each
   3. Perimeter 1 = 6+10+5+5 = 26 cm. Perimeter 2 = 9+15+7.5+7.5 = 39 cm.
   4. Area scale factor = k² = 2.25. Area 2 = 32 × 2.25 = 72 cm²
   5. Ratio of areas = 72/32 = 2.25 = 1.5² = k². ✓

6. Flagpole shadow

   1. Two right triangles: post triangle (height 1.8 m, shadow 2.5 m) and flagpole triangle (height h, shadow 15 m). Both share the same angle of elevation of the sun.
   2. Height corresponds to height; shadow corresponds to shadow.
   3. h/1.8 = 15/2.5 = 6. h = 1.8 × 6 = 10.8 m
   4. Assumption: the sun’s rays are parallel (the sun is very far away), so the angle of elevation of the sun is the same at both locations at the same time.

7. Model aeroplane (scale 1:72)

   1. Real wingspan = 43.5 × 72 = 3132 cm = 31.32 m
   2. Model length = 5832 cm ÷ 72 = 81 cm
   3. Area scale factor = 72² = 5184. Real window area = 2.4 × 5184 = 12 441.6 cm² = 1.244 m²
   4. Volume scale factor = 72³ = 373 248. Real volume = 15 × 373 248 = 5 598 720 cm³ = 5.599 m³

8. DE parallel to BC

   1. ∠A is common. DE ∥ BC, so ∠ADE = ∠ABC (corresponding angles). By AA, ▵ADE ~ ▵ABC.
   2. AD = 5, AB = AD+DB = 5+3 = 8. k = AB/AD = 8/5 = 1.6
   3. BC = DE × k = 6 × 1.6 = 9.6 cm
   4. AC = AE × k = 4 × 1.6 = 6.4. EC = AC − AE = 6.4 − 4 = 2.4 cm
   5. Area ratio = (ADE)/(ABC) = k² for the smaller to larger — but here k is from small to large, so ratio of areas = (AD/AB)² = (5/8)² = 25/64

9. Photo enlargement

   1. k = 30/12 = 2.5
   2. New height = 18 × 2.5 = 45 cm
   3. Original area = 12 × 18 = 216 cm²
   4. Area = 216 × 2.5² = 216 × 6.25 = 1350 cm² (or 30 × 45 = 1350 ✓)
   5. Total dimensions = (30+4) × (45+4) = 34 × 49. Total area = 1666 cm²

10. Find original dimensions from area and scale factor

    1. Width = Area/Length = 180/15 = 12 cm
    2. k (larger to smaller) = 1/2.5 = 0.4
    3. Smaller length = 15 × 0.4 = 6 cm; smaller width = 12 × 0.4 = 4.8 cm. Dimensions: 6 cm × 4.8 cm
    4. Smaller area = 6 × 4.8 = 28.8 cm². Via area scale factor: 180/(2.5²) = 180/6.25 = 28.8 cm². ✓
    5. Perimeter larger = 2(15+12) = 54 cm. Perimeter smaller = 2(6+4.8) = 21.6 cm. Ratio = 21.6/54 = 0.4 = k ✓