Question

Pythagoras theorem
Class 9

Explain following questions with briefly :

1. Biography of Pythagoras
2. Statement of the Pythagoras theorem. 3. Search Internet or library for different proofs of the Pythagoras theorem.
4. Applications of the theorem in daily life. 5. What is a Pythagorean triplet?

Note :
Answer needed to be Contained is

1. All following questions contain atleast 300 words .
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Answer 1

$\underbrace{ \blacksquare \pmb {\sf\: \color{blue}biography \: of \: pythagoras : }}$

➤ Pythagoras was born in Samos, a little island off the western coast of Asia Minor. There is not much information about his life. It is said that he had a good childhood. Growing up with two or three brothers, he was well educated. Though this theorem bears the name of Pythagoras yet the historians are not thoroughly convinced that he developed the first proof of it . They think that Thales who was his teacher , may have first developed the proof of the theorem .

Neugebaur speaks that the Babylonian study and discovery of the diagonal of a square , given the measure of the side as "sufficient proof that Pythagorean theorem was known more than a thousand year before ''. The conception of this theorem may not be entirely Western . According to historians the theorem was known to Hindus. The most abbreviated proof of the theorem was given by Hindu scholar Bhaskara .

$\underbrace{ \blacksquare \pmb {\sf\: \color{blue}statements \: of \: pythagoras \: theorem: }}$

Pythagoras theorem states that ''In a right-angled triangle, the square of the hypotenuse side is equal to the sum of squares of the other two sides''. The sides of this triangle have been named as Perpendicular, Base and Hypotenuse. Here, the hypotenuse is the longest side, as it is opposite to the angle 90°.

$\underbrace{ \blacksquare \pmb {\sf\: \color{blue}proof\: of \: pythagoras \: theorem: }}$

In a right angled triangle the square on the Hypotenuse is equal to the sum of the squares on the he other two sides

Given : A right triangle with leg length a and b and hypotenuse of length c .

Prove : Arrange four copies of he triangle shape. The side of the triangles form two squares .

The area of the outer ∆ is (a+b)² . The area of the inner square is c² . The area of each shaded ∆ is 1/2 ab .

Area of outer square = Area of 4 shaded ∆ + Area of inner square

$\bf: \implies \: ({a + b})^{2} = 4( \frac{1}{2}ab) + {c}^{2}$

$\sf(substituting \: the \: value \: of \: areas \: simplifying)$

$\bf : \implies \: {a}^{2} + 2ab + {b}^{2} = 2ab + {c}^{2}$

$\Rrightarrow \pmb {\bf\: \color{red} {a}^{2} + {b}^{2} = {c}^{2} }$

$\underbrace{ \blacksquare \pmb {\sf\: \color{blue}application \: of \: pythagoras \: theorem: }}$

• Architecture and Construction

• Laying out square angles

• Navigation

• Surveying

• Solving two dimensional real life problems

$\underbrace{ \blacksquare \pmb {\sf\: \color{blue} pythagorean \: triplet: }}$

If a , b and c are such non - zero whole numbers that a² + b² = c² then a ,b and c forms a pythagorean triplet . For eg :- 3, 4 ,5 ; 5,12 ,13 ; etc ..are Pythagorean triplets .

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Answer 2

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