Understand, derive, and apply Thales' Basic Proportionality Theorem and its converse.
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Core rule and equation for the Basic Proportionality Theorem.
When two triangles have all their corresponding angles equal, they are called equiangular triangles.
Imagine looking at a triangular building blueprint versus the actual building. The angles remain identical regardless of scale. This fixed geometric balance ensures that the relative proportions of the sides never change.
Visualize the perpendiculars and cross-lines needed to prove the theorem.
Order the logical steps to prove the Basic Proportionality Theorem using triangle areas.
Arrange the logical steps in correct sequence to complete the proof of the Basic Proportionality Theorem (Theorem 6.1).
Example 1 showing how to derive AD/AB = AE/AC from the BPT.
Problem. If a line intersects sides and of a at and respectively and is parallel to , prove that:
Given. In , with on and on .
Use proportional side lengths to prove that lines are parallel.
Example 2: Prove proportionality on non-parallel sides of a trapezium.
Consider trapezium where and , with line segment intersecting non-parallel sides and at and . We construct diagonal which intersects at point . In , since , applying the Basic Proportionality Theorem gives \frac{AE}{ED} = \text{}. In , since , applying the same theorem gives \frac{CG}{AG} = \text{}, which reciprocates to . By comparing these two equations, we conclude that \frac{AE}{ED} = \text{}.