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.
In one line: Same ratio, same direction.
Practice This ConceptUnderstand it
The converse works backward from ratios to parallelism. If the two sides are divided proportionally, the segment must be parallel to the base.
Formula and rule
If AD/DB = AE/EC, then DE ∥ BC.
See it worked out
Follow the example line by line, then use the same method on new questions.
Worked example
- 1If 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.
- 2So DE ∥ BC.
Method you can reuse
- 1Write the two side ratios in matching order.
- 2Compare the fractions.
- 3If they are equal, conclude the line is parallel.
Sketch it in your answer
Draw triangle ABC with D on AB and E on AC; the ratio equality is the clue for parallelism.
Check yourself
Answer the question in your head before opening the answer.
Try it yourself
What conclusion follows if two sides of a triangle are divided in the same ratio by a line segment?
Think first, then show answerHide answer
Common wrong answer
Saying the line only bisects the triangle because the ratios match. Equal ratios imply parallelism, not necessarily midpoint division.
Don't mix this up
Exam tip
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
- 1
Enough for 1 mark
If a line divides two sides of a triangle in the same ratio, then the line is parallel to the third side.
- 2
Add for 2 marks
Example: If AD/DB = AE/EC, then DE ∥ BC.
- 3
Add for 3 marks
The converse works backward from ratios to parallelism.
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