State and prove basic proportionality theorem.

The Basic Proportionality Theorem, also known as Thales' theorem, is a fundamental concept in geometry that deals with the relationship between parallel lines and transversals. This theorem states that if a line is drawn parallel to one side of a triangle, it divides the other two sides proportionally. Let’s break this down and prove it step by step.

Understanding the Theorem

To visualize this, imagine a triangle ABC. Let’s say we draw a line DE parallel to side BC, intersecting sides AB and AC at points D and E, respectively. According to the Basic Proportionality Theorem, the following relationship holds:

• AD/DB = AE/EC

This means that the ratio of the lengths of the segments created on one side of the triangle (AD and DB) is equal to the ratio of the lengths of the segments on the other side (AE and EC).

Proof of the Theorem

To prove this theorem, we can use the concept of similar triangles. Here’s how the proof unfolds:

1. Consider triangle ABC with DE parallel to BC.
2. By the property of parallel lines, angles are equal. Thus, we have:
• ∠ADE = ∠ABC (corresponding angles)
• ∠AED = ∠ACB (corresponding angles)
3. Since two angles of triangle ADE are equal to two angles of triangle ABC, by the Angle-Angle (AA) similarity criterion, triangle ADE is similar to triangle ABC.
4. From the properties of similar triangles, we know that the ratios of corresponding sides are equal. Therefore, we can write:
• AD/AB = AE/AC
• DB/BC = EC/AC
5. Now, we can express AD and DB in terms of AB and BC:
• AD = AB - DB
• AE = AC - EC
6. By substituting these into our ratios and cross-multiplying, we arrive at:
• AD * EC = AE * DB
7. Rearranging gives us the desired proportionality:
• AD/DB = AE/EC

Applications of the Theorem

The Basic Proportionality Theorem is not just a theoretical concept; it has practical applications in various fields, including architecture, engineering, and even art. For instance, when designing a building, architects often use this theorem to ensure that proportions are maintained for aesthetic and structural integrity.

Real-World Example

Imagine you are creating a scale model of a triangular park. If you want to ensure that the paths within the park maintain the same proportions as the actual park, you can apply the Basic Proportionality Theorem. By drawing paths parallel to the sides of the triangle representing the park, you can determine the lengths of the paths in your model based on the lengths in the actual park.

In summary, the Basic Proportionality Theorem is a powerful tool in geometry that helps us understand the relationships between different parts of a triangle when parallel lines are involved. Its proof through similar triangles not only reinforces the concept but also opens up avenues for practical applications in everyday life.