The same split means parallel

The idea

If a line divides any two sides of a triangle in the same ratio, then the line is parallel to the third side.

Cut the sides, then draw

Mark D on AB so that D cuts AB in the ratio 1 : 2. Mark E on AC so that E cuts AC in the ratio 1 : 2 too.

Now join D and E.

Is the line DE parallel to BC?

• Yes. DE is parallel to BC.
• No. Equal ratios do not make lines parallel.
• We cannot tell without measuring angles.

Find the equal splits

Move D and E on their own: drag them or use the dials. The dashed link between the bars is green when D and E cut their sides in the same ratio. Then look at the line DE.

Same ratio gives parallel

If a line divides two sides of a triangle in the same ratio, AD/DB = AE/EC, then the line is parallel to the third side: DE ∥ BC.

This is the converse of the Basic Proportionality Theorem. The theorem goes from parallel to the same ratio. The converse goes from the same ratio to parallel.

Suppose AD/DB = AE/EC but DE is not parallel to BC. Draw DE′ parallel to BC, with E′ on AC. By the theorem, AD/DB = AE′/E′C. So AE/EC = AE′/E′C. Add 1 to both sides: AC/EC = AC/E′C. So EC = E′C, and E and E′ are the same point. Then DE is the same line as DE′, which is parallel to BC.

Notes

If a line divides two sides of a triangle in the same ratio, AD/DB = AE/EC, then the line is parallel to the third side (converse of the Basic Proportionality Theorem).

Check yourself

In Δ PQR, E is on PQ and F is on PR. PE = 4 cm, EQ = 6 cm, PF = 6 cm and FR = 9 cm. Is EF parallel to QR?

In Δ PQR, E is on PQ and F is on PR. PE = 3 cm, EQ = 5 cm, PF = 4 cm and FR = 6 cm. Is EF parallel to QR?

In Δ ABC, D is on AB with AD = 3 cm and DB = 4 cm. E is on AC with AE = 6 cm. How long must EC be, in cm, so that DE ∥ BC?

Answer: 8

We need AE/EC = AD/DB = 3/4. So 6/EC = 3/4 and EC = 8 cm.

In Δ PQR, S is on PQ and T is on PR with PS/SQ = PT/TR. If ∠PST = 50°, find ∠PQR, in degrees.

Answer: 50

PS/SQ = PT/TR gives ST ∥ QR. Corresponding angles are equal, so ∠PQR = ∠PST = 50°.

• Yes, EF ∥ QR. — correct. Yes! PE/EQ = 4/6 = 2/3 and PF/FR = 6/9 = 2/3. The ratios are equal.
• No, the sides have different lengths.. Equal ratios, not equal lengths, decide it. PE/EQ = 4/6 = 2/3 and PF/FR = 6/9 = 2/3.
• We cannot tell without measuring angles.. No angles are needed. The two ratios are both 2/3, so the line is parallel.
• Yes, EF ∥ QR.. PE/EQ = 3/5 = 0.6 but PF/FR = 4/6 is about 0.67. The ratios are not equal.
• No, EF is not parallel to QR. — correct. Yes! PE/EQ = 3/5 = 0.6 and PF/FR = 4/6 = 2/3, about 0.67. They are not equal.
• We cannot tell without measuring angles.. No angles are needed. The ratios 3/5 and 4/6 are different, so the line is not parallel.