Thales Theorem and Its Application

Thales Theorem, also known as the Basic Proportionality Theorem, states that if a line is drawn parallel to one side of a triangle, it divides the other two sides proportionally. In other words, if a line is drawn parallel to one side of a triangle, it forms several smaller triangles that are similar to the original triangle.

Given Information:
- In the given figure, LM is parallel to CB and LN is parallel to CD.
- We need to prove that AM / AB = AN / AD.

Proof:

Step 1: Establishing the Similarity of Triangles

To prove the given statement, we can start by establishing the similarity of triangles. Let's consider triangle ABC and triangle LAM.

- LM is parallel to CB (given)
- Therefore, by the Thales Theorem, triangle LAM is similar to triangle ABC.

Similarly, considering triangle ACD and triangle LNA:

- LN is parallel to CD (given)
- Hence, by the Thales Theorem, triangle LNA is similar to triangle ACD.

Step 2: Using the Property of Similar Triangles

Since triangles LAM and ABC are similar, we can write the following proportion:

AM / AB = LM / CB ...(1)

Similarly, since triangles LNA and ACD are similar, we have:

AN / AD = LN / CD ...(2)

Step 3: Substituting the Given Information

From the given information, we know that LM is parallel to CB, and LN is parallel to CD. Therefore:

LM / CB = LN / CD

Substituting this equation into equation (1), we get:

AM / AB = LN / CD

Similarly, substituting this equation into equation (2), we have:

AN / AD = LN / CD

Step 4: Conclusion

From equations (3) and (4), we can conclude that AM / AB = AN / AD. Thus, the given statement is proved.

Hence, the ratio of the lengths AM to AB is equal to the ratio of the lengths AN to AD.

Hey see..

given that :- LM//CB

LN//CD

to prove:- AM/AB = AN/AD

proof :- we know that LNM and CBD are straight line

so that . MN//BD

therefore the give figure is a parallelogram so that

MN//LC//BD

so , we know that a Line parallel to one its also parallel to other in a parallelogram

therefore AM =AN

and. AB = AD

so that

AM/AB = AN/AD proved

hope you get helped

thanks