Quick answer: In triangle ABC, if D lies on AB and E on AC with DE parallel to BC, then AD/DB = AE/EC. This is the Basic Proportionality Theorem; similarity also gives AD/AB = AE/AC.
Worked example
AD = 3 cm, DB = 2 cm, AE = 4.5 cm and DE ∥ BC. Find EC.
- Use corresponding split segments: 3/2 = 4.5/EC.
- Cross-multiply: 3EC = 9.
- EC = 3 cm.
- The whole side AC is 4.5 + 3 = 7.5 cm.
Check the result
AD/AB = 3/5 = 0.6 and AE/AC = 4.5/7.5 = 0.6.
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Common mistake to avoid
Do not mix a part-to-part ratio on one side with a part-to-whole ratio on the other. Parallelism is essential.
Try it yourself
1. AD = 4, DB = 6, AE = 2; find EC. 2. If AD/AB = 2/5 and AC = 15, find AE.
Answers and reasoning
1. 4/6 = 2/EC gives EC = 3. 2. AE = (2/5)15 = 6, assuming the same parallel-line setup.
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