A parallel line cuts both sides alike

The idea

Basic Proportionality (Thales) Theorem: a line drawn parallel to one side of a triangle, meeting the other two sides in distinct points, divides those two sides in the same ratio (AD/DB = AE/EC).

A line across a triangle

In triangle ABC the line DE is drawn parallel to the side BC. It cuts AB at D so that AD = 3 and DB = 5.

So D cuts AB in the ratio 3 : 5. Now look at the other side, AC.

In what ratio does E cut AC?

• AE : EC = 3 : 5, the same as on AB
• AE : EC = 1 : 1, E is the middle of AC
• It depends on how long AC is

Slide the line

Drag D or E along the triangle, or use the dial, to slide the line DE. It always stays parallel to BC. The toy measures AD, DB, AE and EC and divides.

The same ratio on both sides

If DE ∥ BC, then AD/DB = AE/EC. A line drawn parallel to one side of a triangle divides the other two sides in the same ratio.

This is the Basic Proportionality Theorem, also called Thales Theorem. On the bars, D and E always cut AB and AC at the same place.

Triangles BDE and CDE have the same base DE and lie between the same parallels DE and BC, so their areas are equal. Triangle ADE has the same height as BDE from E, and the same height as CDE from D. So ar(ADE)/ar(BDE) = AD/DB and ar(ADE)/ar(CDE) = AE/EC. The two areas on the bottom are equal, so AD/DB = AE/EC.

Notes

If DE ∥ BC, then AD/DB = AE/EC. A line drawn parallel to one side of a triangle divides the other two sides in the same ratio (Basic Proportionality Theorem).

Check yourself

In Δ ABC, DE ∥ BC with D on AB and E on AC. AD = 3 cm, DB = 6 cm and AE = 4 cm. Find EC, in cm.

Answer: 8

AD/DB = 3/6 = 1/2, so AE/EC = 1/2 and EC = 2 × 4 = 8 cm.

In Δ ABC, DE ∥ BC, with AD = 2, DB = 6, AE = 3 and EC = 9. Which equation is true?

In Δ ABC, DE ∥ BC. AD = 4 cm, DB = 2 cm and AE = 6 cm. How long is the whole side AC, in cm?

Answer: 9

AD/DB = 4/2 = 2, so AE/EC = 2 and EC = 6 ÷ 2 = 3 cm. AC = AE + EC = 6 + 3 = 9 cm.

In Δ ABC, D is the midpoint of AB and DE ∥ BC. Where is E on AC?

• AD/AB = AE/EC. AD/AB = 2/8 = 1/4 but AE/EC = 3/9 = 1/3. AE/EC compares a part with a part, so it matches AD/DB.
• AD/AB = AE/AC — correct. Yes! AD/AB = 2/8 = 1/4 and AE/AC = 3/12 = 1/4. Part : whole matches part : whole.
• AD/DB = AE/AC. AD/DB = 2/6 = 1/3 but AE/AC = 3/12 = 1/4. AD/DB compares part with part, so it matches AE/EC.
• E is the midpoint of AC — correct. Yes! AD/DB = 1, so AE/EC = 1 and AE = EC.
• E is closer to A than to C. AD = DB, so AD/DB = 1. Then AE/EC = 1 too, and AE = EC.
• It depends on the triangle. Whatever the triangle, AD/DB = 1 forces AE/EC = 1.