Converse of Basic Proportionality Theorem (BPT)
The Converse of the Basic Proportionality Theorem (BPT) or Thales’s Theorem states that if a line intersects two sides of a triangle and divides the sides proportionally, then the line is parallel to the third side of the triangle.
Proof
Let’s prove the converse of the Basic Proportionality Theorem or converse of Thales Theorem is given as follows:
Given: Let us consider a triangle ABC with D and E two points on the side AB and AC such that AD/BD = AE/CE . . .(i).
To Prove: DE∥BC
Construction
To prove the converse of BPT, draw a line DE’∥BC.
Proof
Now, as DE’∥BC in triangle ABC,
Thus, by using the Basic proportionality theorem,
AD/BD = AE’/CE’ . . .(ii)
Using equations (i) and (ii), we get
AE/CE = AE’/CE’
Add 1 to both sides of the equation,
AE/CE + 1 = AE’/CE’ + 1
⇒ (AE+CE)/CE = (AE’+CE’)/CE’
⇒ AC/CE = AC/CE’
⇒ CE = CE’
This can only be possible if E and E’ are coincident.
Thus, E and E’ are the same points.
As, DE∥BC
So, DE∥BC [Hence Proved]
Basic Proportionality Theorem (BPT) Class 10 | Proof and Examples
Basic Proportionality Theorem: Thales theorem is one of the most fundamental theorems in geometry that relates the parts of the length of sides of triangles. The other name of the Thales theorem is the Basic Proportionality Theorem or BPT.
BPT states that if a line is parallel to a side of a triangle that intersects the other sides into two distinct points, then the line divides those sides in proportion.
Let’s learn about the Thales Theorem or Basic Proportionality Theorem in detail, including its statement, proof, and converse as well.
Table of Content
- Basic Proportionality Theorem or Thales Theorem Statement
- Basic Proportionality Theorem Proof
- Construction
- Basic Proportionality Theorem Proof
- Corollary of Thales Theorem
- Articles related to Basic Proportionality Theorem:
- Converse of Basic Proportionality Theorem (BPT)
- Proof
- Construction
- Proof
- Solved Examples on Basic Proportionality Theorem
- Practice Problems on Basic Proportionality Theorem