Basic Proportionality Theorem (Prove) MCQs Quiz | Class 10

Understanding the Basic Proportionality Theorem (BPT)

The Basic Proportionality Theorem, often referred to as Thales Theorem, is a fundamental concept in Euclidean geometry. It establishes a relationship between a line drawn parallel to one side of a triangle and the ratios of the other two sides. Understanding this theorem is crucial for solving problems involving similar triangles and parallel lines.

Basic Proportionality Theorem (BPT)

Statement: If a line is drawn parallel to one side of a triangle to intersect the other two sides in distinct points, then the other two sides are divided in the same ratio.

Consider a triangle ABC. Let a line DE be drawn parallel to the side BC, intersecting AB at D and AC at E. According to the BPT, the ratio of the segments on AB will be equal to the ratio of the segments on AC. That is:

AD / DB = AE / EC

This theorem forms the basis for understanding how parallel lines interact within a triangular structure, leading to proportional segments. The proof typically involves comparing the areas of triangles formed by the construction, often relying on the fact that triangles with the same height and base on the same line have equal areas, or triangles with the same height have areas proportional to their bases.

Converse of Basic Proportionality Theorem

The converse of BPT provides a way to prove that a line is parallel to one side of a triangle.

Statement: If a line divides any two sides of a triangle in the same ratio, then the line is parallel to the third side.

Using the same triangle ABC with points D on AB and E on AC, if we are given that:

AD / DB = AE / EC

Then, by the converse of BPT, we can conclude that the line segment DE is parallel to BC.

Key Applications of BPT and its Converse

• Finding Unknown Lengths: When a line parallel to one side is given, BPT can be used to find the lengths of unknown segments on the other two sides.
• Proving Parallelism: The converse of BPT is essential for proving that a given line segment within a triangle is parallel to its third side by checking the ratios of the other two sides.
• Foundation for Similarity: BPT is closely related to the concept of similar triangles. If DE is parallel to BC, then triangle ADE is similar to triangle ABC.

Quick Revision Points

• Basic Proportionality Theorem (BPT) states: Parallel line to a side -> divides other two sides proportionally.
• Converse of BPT states: Line divides two sides proportionally -> parallel to the third side.
• BPT is also known as Thales Theorem.
• It is a fundamental tool in geometry, especially for problems involving similar triangles and proportional segments.

Practice Questions

Test your understanding with these additional practice questions (no options or answers provided, for self-study):

1. In triangle PQR, a line ST is parallel to QR. If PS = 4 cm, SQ = 6 cm, and PT = 3 cm, find the length of TR.
2. A line intersects sides AB and AC of triangle ABC at D and E respectively. If AD = 5 cm, DB = 10 cm, AE = 3 cm, and EC = 6 cm, demonstrate how you would show that DE is parallel to BC.
3. State the Basic Proportionality Theorem in your own words.
4. If a line segment divides two sides of a triangle in the ratio 1:1, what can you conclude about its relationship to the third side of the triangle?
5. In triangle XYZ, MN is parallel to YZ. If XM/MY = 2/3 and XZ = 15 cm, find the lengths of XN and NZ.