State Basic proportionality theorem?

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BPT:-

If a line is parallel to a side of a triangle which intersects the other sides into two distinct points, then the line divides those sides in proportion.

Answer

Answer

Basic Proportionality Theorem

The Basic Proportionality Theorem, also known as the Thales' Theorem, states that if a line is drawn parallel to one side of a triangle and it intersects the other two sides, then it divides those sides proportionally.

Explanation

Let's consider a triangle ABC, where DE is a line parallel to side BC. The line DE intersects side AB at point F and side AC at point G. According to the Basic Proportionality Theorem:

1. Proportional Segments - The theorem states that the ratio of the lengths of the segments on each side of the triangle is equal. In other words, AF/FB = AG/GC.

2. Similar Triangles - The theorem can also be understood by considering the similar triangles formed. Triangle ADF and triangle ABC are similar, as well as triangle AGE and triangle ABC. This can be proven using the Angle-Angle (AA) similarity criterion.

3. Ratio of Corresponding Sides - Since the triangles are similar, the ratios of corresponding sides are equal. Therefore, AF/AB = DF/BC and AG/AC = EG/BC.

4. Combining Ratios - By substituting the values obtained from the previous step, we can combine the ratios to obtain AF/FB = AG/GC.

5. Proof by Similarity - To further prove the theorem, we can show that the triangles ADF and AGE are similar. This can be done by showing that angle DAF is equal to angle GAE (alternate interior angles) and angle ADF is equal to angle AGE (corresponding angles).

6. Converse of the Theorem - The converse of the Basic Proportionality Theorem is also true. If a line intersects two sides of a triangle such that the segments on each side are divided proportionally, then the line is parallel to the third side.

Example Application

The Basic Proportionality Theorem is often used in various geometric problems and constructions. It helps in solving problems related to similar triangles, finding unknown lengths or ratios, and establishing parallel lines. It is a fundamental concept in geometry that helps in understanding the relationship between parallel lines and triangles.

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