6 min readConcept sheet1-mark MCQ and assertion-reason, 2-mark and 3-mark questions

Converse of BPT and using it

Converse of BPT: if a line divides two sides of a triangle in the same ratio, it is parallel to the third side.

Converse of BPTIn ΔABC, if AD/DB = AE/EC, then DE ∥ BC

In your syllabusStatement only; no proof is asked

BPT vs converseBPT: parallel ⇒ same ratio. Converse: same ratio ⇒ parallel

BPT and its converse

BPT (theorem)Given DE ∥ BC. Then AD/DB = AE/EC. Use it to find a length.

Converse of BPTGiven AD/DB = AE/EC. Then DE ∥ BC. Use it to prove that two lines are parallel.

Read the question. If it gives parallel lines, use BPT. If it asks you to show lines are parallel, use the converse.

Example 1: show that a line is parallel

Problem

In ΔPQR, S is on PQ and T is on PR. PS = 4 cm, PQ = 10 cm, PT = 6 cm and PR = 15 cm. Show that ST ∥ QR.

Working

SQ = PQ − PS = 10 − 4 = 6 cm and TR = PR − PT = 15 − 6 = 9 cm
PS/SQ = 4/6 = 2/3
PT/TR = 6/9 = 2/3
So PS/SQ = PT/TR.
By the converse of BPT, ST ∥ QR.

Answer

ST ∥ QR, because both ratios are 2/3.

Example 2: when the line is not parallel

Problem

In ΔABC, P is on AB and Q is on AC. AP = 2.5 cm, PB = 3.5 cm, AQ = 3 cm and QC = 4 cm. Is PQ ∥ BC?

Working

AP/PB = 2.5/3.5 = 5/7
AQ/QC = 3/4
Cross-multiply to compare: 5 × 4 = 20 but 7 × 3 = 21, so 5/7 ≠ 3/4.
If PQ were parallel to BC, BPT would make these ratios equal. They are not equal, so PQ cannot be parallel to BC.

Answer

No, PQ is not parallel to BC.

Example 3: find a length that makes the line parallel

Problem

In ΔABC, D is on AB and E is on AC. AD = 2.4 cm, DB = 3.6 cm and AE = 3 cm. What must EC be for DE to be parallel to BC?

Working

For DE ∥ BC, the converse of BPT needs AD/DB = AE/EC.
2.4/3.6 = 3/EC
EC = (3 × 3.6)/2.4 = 10.8/2.4 = 4.5 cm

Answer

EC must be 4.5 cm.

How to draw the figure, and tips for BPT problems

Draw ΔABC with the base BC at the bottom. Mark D on AB and E on AC, and join DE.
Write each given length on its side, and mark parallel lines with arrows.
The mid-point theorem (Class 9) fits here. If D and E are mid-points of AB and AC, then AD/DB = AE/EC = 1, so DE ∥ BC by the converse of BPT.
To compare two fractions, cross-multiply: a/b = c/d only if ad = bc.
Write the reason in brackets every time: (by BPT) or (by the converse of BPT).

In real life

A builder can check that a beam is parallel to the floor by measuring. If it cuts both slanting rafters of a roof in the same ratio, it is parallel.

Exam angle

2 marks: “State the converse of the Basic Proportionality Theorem.” If a line divides any two sides of a triangle in the same ratio, then the line is parallel to the third side.

2 marks: “In ΔABC, D and E lie on AB and AC with AD = 3 cm, DB = 4 cm, AE = 4.5 cm and EC = 6 cm. Is DE ∥ BC? Justify.” AD/DB = 3/4 and AE/EC = 4.5/6 = 3/4. The ratios are equal, so DE ∥ BC by the converse of BPT.

Remember it

BPT: parallel → ratio. Converse: ratio → parallel. Same words, arrow reversed.