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medium difficulty5 min

Basic proportionality theorem

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

In one line: Parallel line, proportional cut.

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Understand it

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.

Example

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

Formula and rule

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

See it worked out

Follow the example line by line, then use the same method on new questions.

Worked example

  1. 1In triangle ABC, DE ∥ BC, AD = 4 cm, DB = 6 cm and AE = 5 cm.
  2. 2Using AD/DB = AE/EC, we get 4/6 = 5/EC, so EC = 7.5 cm.

Method you can reuse

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

Sketch it in your answer

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

Check yourself

Answer the question in your head before opening the answer.

Try it yourself

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

Think first, then show answer
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.

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.

Don't mix this up

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

Exam tip

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

Write it in the exam

Each point is worth about one mark. Write point 1 for a 1-mark question and keep adding points as the marks go up.

How CBSE asks it

A numeric question may give a parallel line inside a triangle and ask for one missing segment.
  1. 1

    Enough for 1 mark

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

  2. 2

    Add for 2 marks

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

  3. 3

    Add for 3 marks

    This theorem is a ratio rule inside a triangle.

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