pythagoras theorem

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… and its proof! Pythagoras Theorem……

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Pythagoras lived in the 500s BC, and was one of the first Greek mathematical thinkers. He spent most of his life in the Greek colonies in Sicily and southern Italy. He had a group of followers called The Pythagoreans. Pythagoreans were interested in philosophy, but especially in music and mathematics, two ways of making order out of chaos. Pythagoras himself is best known for proving that the Pythagorean Theorem was true. The Sumerians, two thousand years earlier, already knew that it was generally true, and they used it in their measurements, but Pythagoras is said to have proved that it would always be true. Pythagoras

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The famous theorem by Pythagoras defined the relationship between the three sides of a right triangle. Pythagorean Theorem says that in a right triangle, the sum of the squares of the two right-angle sides will always be the same as the square of the hypotenuse. Statement of Pythagoras Theorem

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Now, we shall prove this theorem using the concept of similarity of triangles. In proving this, we shall make use of a result related to similarity of two triangles formed by the perpendicular to the hypotenuse from the opposite vertex of the right triangle.

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Now, let us take a right triangle ABC, right angled at B. Let BD be the perpendicular to the hypotenuse AC . A B C D You may note that in Δ ADB and Δ ABC ∠ A = ∠ A (Common) and ∠ ADB = ∠ ABC (Right) So, Δ ADB ~ Δ ABC (AA Similarity Criteria) (1) Similarly, Δ BDC ~ Δ ABC (AA Similarity Criteria) (2)

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So, from (1) and (2), triangles on both sides of the perpendicular BD are similar to the whole triangle ABC. Also, since Δ ADB ~ Δ ABC and Δ BDC ~ Δ ABC So, Δ ADB ~ Δ BDC The above discussion leads to the following theorem : Theorem 1 : If a perpendicular is drawn from the vertex of the right angle of a right triangle to the hypotenuse then triangles on both sides of the perpendicular are similar to the whole triangle and to each other.

Let us now apply this theorem in proving the Pythagoras Theorem:: 

Let us now apply this theorem in proving the Pythagoras Theorem: Pythagoras Theorem : In a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.

Proof……: 

Proof…… We are given a right triangle ABC right angled at B. We need to prove that AC 2 = AB 2 + BC 2 Let us draw BD ⊥ AC (see Fig). Now, Δ ADB ~ Δ ABC (Theorem 1) So, AD/AB = AB/AC (Sides are proportional) Or, AD.AC = AB 2 (1) A B C D

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Also, Δ BDC ~ Δ ABC (Theorem 1) So, CD/BC = BC/AC Or, CD.AC = BC 2 (2) Adding (1) and (2), AD.AC + CD.AC = AB 2 + BC 2 or, AC (AD + CD) = AB 2 + BC 2 or, AC.AC = AB 2 + BC 2 or, AC 2 = AB 2 + BC 2 Hence Proved.

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The above theorem was earlier given by an ancient Indian mathematician Baudhayan (about 800 B.C.) in the following form : The diagonal of a rectangle produces by itself the same area as produced by its both sides (i.e., length and breadth). For this reason, this theorem is sometimes also referred to as the Baudhayan Theorem.

This Project has been compiled and presented by :: 

This Project has been compiled and presented by : Class : Tenth “D” Roll : 10431 Shashank . S

In the Supervision of :: 

In the Supervision of : Our Respected Mathematics Teacher Class : Tenth “D” Mrs. Agasthi

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That’s All Folks!