Solving Problems with Similar Triangles

Concept Overview

• When given two similar triangles, sides that correspond to each other are in proportion.
• To find a missing side in similar triangles, set up a proportion and solve for the unknown value using cross multiplication.

Example 1: Finding Missing Side

• Triangles: Triangle ABC and Triangle DEF
• Given Values:
  • AB = 8, BC = 6, DE = 12, EF = x
  • Angle A ≈ Angle D, Angle C ≈ Angle F
• Steps:
  1. Set up the proportion: [\frac{AB}{DE} = \frac{BC}{EF} \Rightarrow \frac{8}{12} = \frac{6}{x}]
  2. Cross multiply: [8x = 6 \times 12 = 72]
  3. Solve for x: [x = \frac{72}{8} = 9]
• Solution: EF = 9

Example 2: Finding Multiple Missing Sides

• Triangles: Triangle ABC and Triangle DEF

• Given Values:

  • AB = 18, BC = 24, DE = 30, EF = x, DF = 40
  • Angle A ≈ Angle F, Angle B ≈ Angle E, Angle C ≈ Angle D

• Steps to Find x:

  1. Set up the proportion: [\frac{BC}{DE} = \frac{AB}{EF} \Rightarrow \frac{24}{30} = \frac{18}{x}]
  2. Cross multiply and simplify: [24x = 18 \times 30]
     • Break down coefficients to cancel out common factors.
     • [24 = 6 \times 4, 30 = 6 \times 5, 18 = 9 \times 2]
     • Cancel common factors: [9 \times 5 = 45, 45 = 2x \Rightarrow x = \frac{45}{2} = 22.5]
  • Solution: EF = 22.5

• Steps to Find y:

  1. Set up the proportion: [\frac{BC}{DE} = \frac{y}{DF} \Rightarrow \frac{24}{30} = \frac{y}{40}]
  2. Cross multiply and simplify: [30y = 24 \times 40]
     • Breakdown and cancellation: [30 = 3 \times 10, 24 = 3 \times 8, 40 = 4 \times 10]
     • Cancel common factors: [y = 8 \times 4 = 32]
  • Solution: y = 32

Practice Example

• Triangles: Triangle ABC and Triangle DEF
• Given Values:
  • AB = 15, BC = 9, EF = x, DF = x + 10
  • Angle A ≈ Angle D, Angle B ≈ Angle E, Angle C ≈ Angle F
• Steps:
  1. Set up the proportion: [\frac{AB}{DF} = \frac{BC}{EF} \Rightarrow \frac{15}{x+10} = \frac{9}{x}]
  2. Cross multiply: [15x = 9(x + 10)]
  3. Expand and solve: [15x = 9x + 90]
     • [6x = 90 \Rightarrow x = 15]
• Solution: x = 15
  • DF = 25 (since 15 + 10)

Final Example

• Triangles: Triangle ABC and Triangle DEC
• Given Values:
  • AB = 8, AC = 5, BC = 7, DC = x, DE = 12
  • Angle A ≈ Angle E, Angle B ≈ Angle D, Angle C ≈ Angle F
• Steps:
  1. Set up proportion: [\frac{AB}{DE} = \frac{BC}{DC} \Rightarrow \frac{8}{12} = \frac{7}{x}]
  2. Cross multiply and simplify: [8x = 7 \times 12]
     • Breakdown and cancellation: [8 = 4 \times 2, 12 = 4 \times 3]
       • Cancel common factors: [2x = 7 \times 3 \Rightarrow x = \frac{21}{2} = 10.5]
• Solution: DC = 10.5

Conclusion

• Understanding and identifying corresponding angles and sides in similar triangles is crucial for setting up correct proportions.
• Cross multiplying and simplifying helps in solving the equations effectively.