Pythagoras Theorem Tips,Formula ,Proof and its application with examples

The Pythagoras theorem is a crucial idea that is applied while resolving mathematical problems. The relationship between a triangle's sides is explained by the theorem. Pythagoras, a Greek philosopher who lived in the sixth century B.C., derived the Pythagoras theorem and declared it to be a fundamental characteristic of right-angled triangles. Thus, the property bears his name. Any right-angled triangle that possesses the Pythagoras property must be one. Let's examine the Pythagoras theorem's assertion, formulations, justification, uses, and examples.If you are preparing for competitive exams and looking for expert guidance, you can download our General Knowledge Free Ebook Download Now.

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Table of content

• Pythagoras Theorem
• Proof for Pythagoras' Theorem
• Applications of the Pythagoras Theorem

 

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Pythagoras Theorem

The hypotenuse of a right-angled triangle is the side that is directly across from the right angle, while the other two sides are referred to as the triangle's legs. The base and perpendicular sides are the other two, with the hypotenuse being the longest.

In a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides, according to Pythagoras's theorem.



The Pythagoras theorem formula can be derived as follows from the preceding figure:
AB² + BC² = AC²

Proof for Pythagoras' Theorem

An ABC right triangle with a B right angle is shown to us. Assume that BD is parallel to the hypotenuse AC. You need to be aware of the statement "Triangles on either side of a perpendicular are comparable to the complete triangle and to each other if the perpendicular is drawn from the vertex of the right angle of a right triangle to the hypotenuse."
 

Let us draw BD ⊥ AC (See Fig)

∠ A = ∠ A
and ∠ ADB = ∠ ABC
So, ∆ ADB ~ ∆ ABC
Similarly, ∆ BDC ~ ∆ ABC
Now, ∆ ADB ~ ∆ ABC

So, AD/AB = AB/AC (Sides are proportional)

or, AD. AC = AB²  (1)
Also, ∆ BDC ~ ∆ ABC

So, CD/BC = BC/AC or CD . AC = BC²   (2)

Adding (1) and (2),

AD. AC + CD . AC = AB² + BC²
or, AC (AD + CD) = AB² + BC²
or, AC. AC = AB² + BC²

or, AC² = AB² + BC²

Applications of the Pythagoras Theorem

The following are some examples of Pythagoras' Theorem applications: 

• to determine whether a triangle is a right-angled triangle or not, as well as to solve triangle-based problems. 
• to figure out a square's diagonal. 
• Architecture, woodworking, and other types of physical construction apply this theorem.

For example :

 ∆ ABC is right-angled at C. If AC = 5 cm and BC = 12 cm find the length of AB.

Refer to the figure given on the right.

As the triangle is right-angled, by Pythagoras theorem,
AB² = AC² + BC²

AB² = 5² + 12²
AB² = 25 + 144
AB² = 169 = 13²

Hence, AB = 13 cm

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Since both triangles' sides are the same lengths a, b and c, the triangles are congruent and must have the same angles. Therefore, the angle between the side of lengths a and b in the original triangle is a right angle. The above proof of the converse makes use of the Pythagorean theorem itself.