Concept 6.2
The Basic Proportionality Theorem
Source: NCERT Mathematics, Class 1010 minute lesson
Formal statement
Basic Proportionality Theorem: if a line is drawn parallel to one side of a triangle to intersect the other two sides in distinct points, the other two sides are divided in the same ratio. In △ABC with DE || BC, AD/DB = AE/EC.
The intuition
A line parallel to the base cuts off a smaller triangle of the same shape at the top. So it slices both sides at the same fraction of the way down.
The ratio compares the two pieces of each side: top piece to bottom piece. It does not say DE/BC = AD/DB; that would mix a piece with the whole.
The attention
Look at the pieces of the two cut sides: top against bottom.
DE || BC ⇒ AD/DB = AE/EC not DE/BC = AD/DB
Look hereNot here
- Look here
- The pieces of the two cut sides. The theorem compares the two pieces of AB with the two pieces of AC. The cut is at the same fraction of the way down on both sides.
- Not here
- Comparing DE with BC. DE/BC compares the small triangle with the big one. That needs AD/AB, the top piece over the whole side, not AD/DB.
- What it teaches
- When a ratio appears, say out loud which two lengths it compares.