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Converse of basic proportionality theorem

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

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Student-friendly explanation

The converse works backward from ratios to parallelism. If the two sides are divided proportionally, the segment must be parallel to the base.

How to write this in exams

  1. 1

    Start with the exact idea

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

  2. 2

    Then show how to use it

    1. Write the two side ratios in matching order. 2. Compare the fractions. 3. If they are equal, conclude the line is parallel.

  3. 3

    Add one concrete example

    If AD/DB = AE/EC, then DE ∥ BC.

  4. 4

    Avoid this incomplete answer

    Saying the line only bisects the triangle because the ratios match. Equal ratios imply parallelism, not necessarily midpoint division.

Definition

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

Example

If AD/DB = AE/EC, then DE ∥ BC.

Rule to remember

If AD/DB = AE/EC, then DE ∥ BC.

Memory hook

Same ratio, same direction.

Examples and method

Worked example

If AD = 2 cm, DB = 3 cm, AE = 4 cm and EC = 6 cm, then AD/DB = 2/3 and AE/EC = 4/6 = 2/3. So DE ∥ BC.

Method to apply

1. Write the two side ratios in matching order. 2. Compare the fractions. 3. If they are equal, conclude the line is parallel.

Diagram support

Draw triangle ABC with D on AB and E on AC; the ratio equality is the clue for parallelism.

How CBSE asks it

A question may give side divisions and ask whether the line is parallel to the third side.

Avoid common mistakes

Common confusion

Students think the line is parallel only after measuring angles. The ratio test itself is enough.

Common wrong answer

Saying the line only bisects the triangle because the ratios match. Equal ratios imply parallelism, not necessarily midpoint division.

Exam tip

Look for equal ratios on the two sides that are cut by the same line.

Quick check

What conclusion follows if two sides of a triangle are divided in the same ratio by a line segment?

The line segment is parallel to the third side by the converse of the Basic Proportionality Theorem. Equal ratios are the clue that the line is parallel.

Answer writing and exam use

1-mark answer

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

2-mark answer

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

3-mark answer

The converse works backward from ratios to parallelism. If the two sides are divided proportionally, the segment must be parallel to the base. If AD/DB = AE/EC, then DE ∥ BC. If AD = 2 cm, DB = 3 cm, AE = 4 cm and EC = 6 cm, then AD/DB = 2/3 and AE/EC = 4/6 = 2/3. So DE ∥ BC. A question may give side divisions and ask whether the line is parallel to the third side. Saying the line only bisects the triangle because the ratios match. Equal ratios imply parallelism, not necessarily midpoint division.
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