State and prove Basic Proportionality Theorem.
State and prove Basic Proportionality Theorem.
Statement: If a line is drawn parallel to one side of a triangle intersecting the other two sides at distinct points, the other two sides are divided in the same ratio — i.e., if in with D on AB and E on AC, then . Proved below using the ratio of triangle areas with equal heights.
Marking Scheme
- 11 mark: correct statement of the theorem with proper labelling (D on AB, E on AC, ).
- 21 mark: correct 'Given', 'To Prove' and construction (joining BE, CD and drawing , ).
- 31 mark: correctly deriving .
- 41 mark: correctly deriving .
- 51 mark: correctly justifying (same base DE, same parallels DE and BC) and concluding .
Hint
Join BE and CD, drop perpendiculars and , and compare triangle areas using the 'same base, same parallels ⟹ equal area' property for triangles DBE and DEC.
Quick Oral Answer
Basic Proportionality Theorem states that a line parallel to one side of a triangle divides the other two sides in the same ratio; it is proved by comparing the areas of triangles ADE, DBE and DEC, using the fact that DBE and DEC have equal area since they share base DE and lie between the same parallels DE and BC.
Analysis & Explanation
The Basic Proportionality Theorem (Thales' Theorem) is one of the most frequently tested proofs in Class 10 Geometry, underpinning the entire similar-triangles chapter.
Proof strategy
- Uses triangle areas as a bridge: triangles with equal height have areas proportional to their bases.
- Construction of two perpendiculars (, ) is essential to compare areas.
- The unifying step is that , since both lie on the same base DE and between the same parallels DE and BC.
Common mistakes
- Omitting the construction of EN and DM explicitly, even if the final ratio is correct.
- Asserting without justifying it via "same base, same parallels."
Real-world/foundational connection
- BPT underlies similarity criteria (AA/SAS/SSS), the Pythagoras theorem proof, and practical scaling problems like maps, blueprints, and shadow-based height calculations.
Common Mistakes
- 1Stating the theorem's conclusion without properly labelling which side is divided by D and which by E.
- 2Omitting the construction of perpendiculars EN and DM, jumping straight to the area ratio without justification.
- 3Failing to justify why ar(DBE) = ar(DEC), merely asserting it without invoking 'triangles on the same base and between the same parallels are equal in area.'
Interesting Facts
The Basic Proportionality Theorem is attributed to the ancient Greek mathematician Thales of Miletus (circa 624–546 BCE), who is said to have used similar-triangle reasoning to measure the height of the Great Pyramid of Giza using shadows.
BPT is the direct foundation for deriving the Pythagoras theorem using similar triangles formed by the altitude to the hypotenuse in a right triangle — a proof method included in the NCERT Class 10 textbook.
The converse of BPT (if a line divides two sides of a triangle in the same ratio, it is parallel to the third side) is equally examinable and is used to prove that a line is parallel without measuring angles directly.
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Frequently Asked Questions
Why is BPT also called Thales' Theorem?
It is named after the ancient Greek mathematician Thales of Miletus, who is credited with first using this proportionality property of parallel lines and triangles.
Why do we need to draw EN and DM in the proof?
These perpendiculars serve as the common heights needed to express the areas of triangles ADE, DBE and ADE, DEC in terms of their respective bases AD, DB and AE, EC, which is the key step in relating the area ratios to the side ratios.
What is the converse of BPT and is it also examinable?
Yes; the converse states that if a line divides two sides of a triangle in the same ratio, then it is parallel to the third side, and it is commonly tested in CBSE numerical and proof-based questions.