From a parallel to a similarity
The parallel line does more than split two sides. It also gives two equal angles.
- Triangle ABC, with DE parallel to BC.
- Angle A is shared by both triangles.
- Angle ADE equals angle ABC.
- AD/AB = AE/AC = DE/BC = 2/5.
The corner triangle
Let DE be parallel to BC. Triangles ADE and ABC share the angle at A. AB crosses the two parallel lines, DE and BC. So ∠ADE and ∠ABC are corresponding angles. They sit in the same place at each line. Corresponding angles on parallel lines are equal.
Two angles match. By AA, triangle ADE is similar to triangle ABC.
Similar triangles have one scale factor. Match the corners in order. A pairs with A, D with B, E with C. So AD ÷ AB = AE ÷ AC = DE ÷ BC.
DE is parallel to BC. So triangle ADE is similar to triangle ABC. Then AD ÷ AB = AE ÷ AC = DE ÷ BC. Use the whole side AB, not the piece DB.
An example
AD = 4 and DB = 6, so AB = 10. AE = 6 and EC = 9, so AC = 15. The factor is 4/10 = 2/5. The factor also equals AE ÷ AC = 6/15 = 2/5. So DE = 2/5 × BC.
| Ratio | Value |
|---|---|
| AD / AB | 4/10 = 2/5 |
| AE / AC | 6/15 = 2/5 |
| DE / BC | 2/5 |
Slide D until DE is the length asked. Read the working below the slider.
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