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Basic proportionality theorem

A line drawn parallel to one side of a triangle divides the other two sides in the same ratio.

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Student-friendly explanation

This theorem is a ratio rule inside a triangle. The parallel line creates smaller triangles that share angles with the bigger triangle, so the cut sides stay proportional.

How to write this in exams

  1. 1

    Start with the exact idea

    A line drawn parallel to one side of a triangle divides the other two sides in the same ratio.

  2. 2

    Then show how to use it

    1. Identify the side parallel to the third side. 2. Write the proportionality relation. 3. Substitute the given lengths. 4. Solve the missing value.

  3. 3

    Add one concrete example

    If DE ∥ BC in triangle ABC, then AD/DB = AE/EC.

  4. 4

    Avoid this incomplete answer

    Using the lengths on the same side without matching the cut segments. The theorem compares the two sides divided by the parallel line.

Definition

A line drawn parallel to one side of a triangle divides the other two sides in the same ratio.

Example

If DE ∥ BC in triangle ABC, then AD/DB = AE/EC.

Rule to remember

If DE ∥ BC in triangle ABC, then AD/DB = AE/EC.

Memory hook

Parallel line, proportional cut.

Examples and method

Worked example

In triangle ABC, DE ∥ BC, AD = 4 cm, DB = 6 cm and AE = 5 cm. Using AD/DB = AE/EC, we get 4/6 = 5/EC, so EC = 7.5 cm.

Method to apply

1. Identify the side parallel to the third side. 2. Write the proportionality relation. 3. Substitute the given lengths. 4. Solve the missing value.

Diagram support

Draw triangle ABC, place D on AB and E on AC, and mark DE parallel to BC.

How CBSE asks it

A numeric question may give a parallel line inside a triangle and ask for one missing segment.

Avoid common mistakes

Common confusion

Students often write the side ratio in the wrong order or forget that the line must be parallel.

Common wrong answer

Using the lengths on the same side without matching the cut segments. The theorem compares the two sides divided by the parallel line.

Exam tip

First check the parallel mark, then write the ratio of the two cut sides carefully.

Quick check

When a line is parallel to one side of a triangle, what happens to the other two sides?

The line divides the other two sides proportionally. That means the two cut sides are divided in the same ratio whenever the line is parallel to the third side.

Answer writing and exam use

1-mark answer

A line drawn parallel to one side of a triangle divides the other two sides in the same ratio.

2-mark answer

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

3-mark answer

This theorem is a ratio rule inside a triangle. The parallel line creates smaller triangles that share angles with the bigger triangle, so the cut sides stay proportional. If DE ∥ BC in triangle ABC, then AD/DB = AE/EC. In triangle ABC, DE ∥ BC, AD = 4 cm, DB = 6 cm and AE = 5 cm. Using AD/DB = AE/EC, we get 4/6 = 5/EC, so EC = 7.5 cm. A numeric question may give a parallel line inside a triangle and ask for one missing segment. Using the lengths on the same side without matching the cut segments. The theorem compares the two sides divided by the parallel line.
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