What is the BPT theorem for Class 10?

The BPT Theorem also known as Thales' Theorem. This theorem is a key concept in geometry, particularly in the study of similar triangles. Here's a detailed explanation:

Statement of the Theorem:

If a line is drawn parallel to one side of a triangle and it intersects the other two sides, then the two sides are divided proportionally.

In simpler terms:

• In a triangle, if a line is drawn parallel to one of its sides, cutting the other two sides, the segments of those two sides are proportional.

Mathematical Representation:

If in triangle ABC, a line DE is drawn parallel to side BC and intersects the other two sides AB and AC at points D and E respectively, then:

(AD/DB) = (AE/EC)

Where:

• AD and DB are parts of side AB
• AE and EC are parts of side AC

Proof (Conceptual):

This theorem can be proven using similar triangles. Since the line DE is parallel to side BC, triangles ADE and ABC are similar by the Basic Proportionality Criterion or AA (Angle-Angle) Similarity Criterion. As a result, the corresponding sides of these triangles are proportional.

Application of the BPT:

• Area Calculations: The BPT is often used to determine the proportionality of areas in geometric figures.
• Solving Problems: It is useful in problems involving proportional parts of triangles when a line is parallel to one side.