Pythagorean Theorem PDF

Pythagorean Theorem PDF Details
PDF Name	Pythagorean Theorem PDF
No. of Pages	34
PDF Size	1.52 MB
Language	English
Category	English
Source	mathcentre.ac.uk
Download Link	Available ✔
Downloads	17

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Pythagorean Theorem

Dear readers, here we are providing the Pythagorean Theorem PDF to all of you. Pythagoras’ theorem is a very popular and one of the most important theorems. It can be proved in multiple ways There are various formulas that you can use to solve it and many students solve it differently to reach the same answer.

One of the most common solving methods is the algebraic method and the other popular one is the similar triangles method. If you want to understand and learn both methods then you should learn them individually in order to understand the proof of this theorem. Through this pdf, You will learn theorem and use it to solve problems involving right-angled triangles.

You will also learn a less-familiar form of the theorem. If you want to get an expert in solving techniques that are explained in the following pdf then you should undertake plenty of practice exercises so that they become second nature. The Theorem of Pythagoras is a well-known theorem.

Pythagorean Theorem Formula PDF

If c denotes the length of the hypotenuse and a and b denote the two lengths of the legs of a right triangle, then the Pythagorean theorem can be expressed as the Pythagorean equation:

{\displaystyle a^{2}+b^{2}=c^{2}.}

If only the lengths of the legs of the right triangle are known but not the hypotenuse, then the length of the hypotenuse can be calculated with the equation

{\displaystyle c={\sqrt {a^{2}+b^{2}}}.}

If the length of the hypotenuse and of one leg is known, then the length of the other leg can be calculated as

{\displaystyle a={\sqrt {c^{2}-b^{2}}}}

or

{\displaystyle b={\sqrt {c^{2}-a^{2}}}.}

A generalization of this theorem is the law of cosines, which allows the computation of the length of any side of any triangle, given the lengths of the other two sides and the angle between them. If the angle between the other sides is a right angle, the law of cosines reduces to the Pythagorean equation.

Pythagorean Theorem PDF Algebraic Proofs

The theorem can be proved algebraically using four copies of the same triangle arranged symmetrically around a square with side c, as shown in the lower part of the diagram.^[5] This results in a larger square, with side a + b and area (a + b)². The four triangles and the square side c must have the same area as the larger square,

{\displaystyle (b+a)^{2}=c^{2}+4{\frac {ab}{2}}=c^{2}+2ab,}

giving

{\displaystyle c^{2}=(b+a)^{2}-2ab=b^{2}+2ab+a^{2}-2ab=a^{2}+b^{2}.}

A similar proof uses four copies of a right triangle with sides a, b and c, arranged inside a square with side c as in the top half of the diagram.^[6] The triangles are similar with area {\displaystyle {\tfrac {1}{2}}ab}, while the small square has side b − a and area (b − a)². The area of the large square is therefore

{\displaystyle (b-a)^{2}+4{\frac {ab}{2}}=(b-a)^{2}+2ab=b^{2}-2ab+a^{2}+2ab=a^{2}+b^{2}.}

But this is a square with side c and area c², so

{\displaystyle c^{2}=a^{2}+b^{2}.}

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