The basic proportionality theorem
In triangle ABC, AB is 10 cm long. Slide a line DE parallel to BC up and down, and change the length of AC: AD ÷ DB always equals AE ÷ EC.
Core concept
If a line parallel to one side of a triangle cuts the other two sides at D and E, it divides them in the same ratio: AD/DB = AE/EC (the basic proportionality, or Thales, theorem)
Topic: Triangles: similarity
Draw a triangle ABC and a line DE parallel to BC, with D on AB and E on AC. The basic proportionality theorem (also called Thales' theorem) says the line cuts the two sides in the same ratio: A way to compare two amounts, like 2 cups of rice to 3 cups of water (2 : 3). All key words: AD ÷ DB = AE ÷ EC. Here AB = 10 cm, so if AD = 4 cm then DB = 6 cm and the ratio is 4 : 6. On AC the same ratio holds, so AE = AC × 4 ÷ 10. The theorem is the key to similar triangles, and its converse says that a line cutting two sides in the same ratio must be parallel to the third side.
What you will learn
- State the basic proportionality theorem.
- Show that AD/DB = AE/EC for different positions of DE and lengths of AC.
- Use the theorem to find a missing length.
Before you start
- Know ratio and proportion, and what parallel lines are.
Materials
- Paper, a ruler, a pencil and a set square
Equipment
- A calculator
Procedure
- Record DE with AD = 4 cm and AC = 15 cm.
- Move DE to AD = 5 cm and record.
- Change AC to 8 cm and record.
- Move DE to AD = 8 cm with AC = 12 cm and record.
How this simulation is simplified
- AB is 10 cm long and DE is always parallel to BC.