Welcome to our explanation of the Basic Proportionality Theorem, also known as Thales Theorem. This fundamental theorem in geometry states that if a line is drawn parallel to one side of a triangle, it divides the other two sides proportionally. Let's visualize this with triangle ABC where line DE is parallel to side BC.
Now let's understand how the proportions work. When line DE is parallel to BC, it creates segments on both sides AB and AC. On side AB, we have segments AD and DB. On side AC, we have segments AE and EC. The theorem tells us that the ratio of AD to DB equals the ratio of AE to EC. This means the parallel line divides both sides in exactly the same proportion.
Let's verify the theorem with a numerical example. Suppose AD equals 3 units and DB equals 2 units. On the other side, AE equals 6 units and EC equals 4 units. Now we calculate the ratios. AD divided by DB equals 3 divided by 2, which equals 1.5. AE divided by EC equals 6 divided by 4, which also equals 1.5. Since both ratios are equal, this confirms that the Basic Proportionality Theorem holds true.
The Basic Proportionality Theorem also has a converse, which is equally important. The converse states that if a line divides two sides of a triangle in the same ratio, then that line must be parallel to the third side. Mathematically, if AD divided by DB equals AE divided by EC, then line DE is parallel to BC. This converse is very useful in geometry for proving that two lines are parallel when we know the ratios are equal.
To summarize what we have learned about the Basic Proportionality Theorem: it states that a line parallel to one side of a triangle divides the other two sides proportionally. The mathematical relationship shows that AD divided by DB equals AE divided by EC. The converse theorem helps us prove that lines are parallel when we know the ratios are equal. This fundamental theorem is essential in geometric similarity and many mathematical proofs.