BASIC PROPTIONALITY THEOREM (THALES THEOREM)

a brief of thales theorem for class 10 mathematics review.

1. BASIC PROPORTIONALITY THEOREM AND CONVERSE OF IT!!!!!!

2. What is basic proportionality theorem? This theorem is also called Thales theorem. It states that, “ If a line is drawn parallel to one side of a triangle intersecting other two sides, then it divides the two sides in the same ratio.”

3. Proof of Thale’s Theorem Given : In ∆ABC , DE || BC and intersects AB in D and AC in E. To Prove : AD / DB = AE / EC Construction : Join BC,CD and draw EF ┴ BA and DG ┴ CA Proof: Statements Reasons 1) EF ┴ BA 1) Construction 2) EF is the height of ∆ADE and ∆DBE 2) Definition of perpendicular 3)Area(∆ADE) = (AD .EF)/2 3)Area = (Base .height)/2 4)Area(∆DBE) =(DB.EF)/2 4) Area = (Base .height)/2 5)(Area(∆ADE))/(Area(∆DBE)) = AD/DB 5) Divide (4) by (5) 6) (Area(∆ADE))/(Area(∆DEC)) = AE/EC 6) Same as above 7) ∆DBE ~∆DEC 7) Both the ∆s are on the same base and between the same || lines. 8) Area(∆DBE)=area(∆DEC) 8) If the two triangles are similar their areas are equal 9) AD/DB =AE/EC 9) From (5) and (6) and (7)

4. History of Thales The ancient Greek philosopher Thales was born in Miletus in Greek Ionia. Aristotle, the major source for Thales's philosophy and science, identified Thales as the first person to investigate the basic principles, the question of the originating substances of matter and, therefore, as the founder of the school of natural philosophy. Thales was interested in almost everything, investigating almost all areas of knowledge, philosophy, history, science, mathematics, engineering, geography, and politics. He proposed theories to explain many of the events of nature, the primary substance, the support of the earth, and the cause of change. Thales was much involved in the problems of astronomy and provided a number of explanations of cosmological events which traditionally involved supernatural entities. His questioning approach to the understanding of heavenly phenomena was the beginning of Greek astronomy. Thales' hypotheses were new and bold, and in freeing phenomena from godly intervention, he paved the way towards scientific endeavor. He founded the Milesian school of natural philosophy, developed the scientific method, and initiated the first western enlightenment. A number of anecdotes is closely connected to Thales' investigations of the cosmos. When considered in association with his hypotheses they take on added meaning and are most enlightening. Thales was highly esteemed in ancient times, and a letter cited by Diogenes Laertius, and purporting to be from Anaximenes to Pythagoras, advised that all our discourse should begin with a reference to Thales (D.L. II.4).

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

6. PROOF OF CONVERSE Given : A Δ ABC and a line intersecting AB in D and AC in E, such that AD / DB = AE / EC. Prove that : DE || BC Proof: Statements Reasons 1) DF || BC 1) By assumption 2) AD / DB = AF / FC 2) By Basic Proportionality theorem 3) AD / DB = AE /EC 3) Given 4) AF / FC = AE / EC 4) By transitivity (from 2 and 3) 5) (AF/FC) + 1 = (AE/EC) + 1 5) Adding 1 to both side 6) (AF + FC )/FC = (AE + EC)/EC 6) By simplifying 7) AC /FC = AC / EC 7) AC = AF + FC and AC = AE + EC 8) FC = EC 8) As the numerator are same so denominators are equal Hence DE II BC

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9. CREDITS AKSHAT RAAHUL JAYESH JADHAV KAUSHIK PRAYAGA ANUSHKA NAIR MEGHNA SAXENA

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