Drawing a line parallel to a side of a triangle divides the sides proportionally.
Triangle and Parallel Line Theorem
In △ABC, when DE ∥ BC (D lies on AB and E lies on AC):
AD:DB = AE:EC = AD:AB = AE:AC
📘 Example ①
In △ABC, when DE ∥ BC, AD=4, DB=2, and AE=6, find EC.
Answer:From AD:DB = AE:EC, 4:2 = 6:EC → EC = 6 × 2/4 = 3
In △ABC, when DE ∥ BC, AD=4, DB=2, and AE=6, find EC.
Answer:From AD:DB = AE:EC, 4:2 = 6:EC → EC = 6 × 2/4 = 3
📘 Example ②
When AD:AB = 3:5 and AB=10, find DB.
Answer:AD = 10 × 3/5 = 6. DB = 10 - 6 = 4
When AD:AB = 3:5 and AB=10, find DB.
Answer:AD = 10 × 3/5 = 6. DB = 10 - 6 = 4
💡 Key Point
- If lines are parallel, the ratios are equal: AD:DB = AE:EC
- The converse also holds: if AD:DB = AE:EC, then DE ∥ BC
Practice Problems
- In △ABC, when DE ∥ BC, AD=3, and DB=5, find AD:AB.
- In the problem above, find EC when AE=6.
Answer and Explanation
- Answer:3:8. AD:AB = 3:(3+5) = 3:8.
- Answer:10. AD:DB = AE:EC → 3:5 = 6:EC → EC=10.