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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.Current Affairs Ebook Free PDF: Download Here
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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
Also check,
How is Pythagoras theorem proved?
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.Why is Pythagoras theorem true?
It's easy to see from the fact that angles in a triangle add up to 180◦ that it is actually a square). There are also four right triangles with base a and height b. The conclusion is that a2 + b2 = c2, which is the Pythagorean Theorem.
Is Pythagoras theorem valid?
The mathematical proof of the Pythagorean theorem does not depend on the geometry of the space of the world that we inhabit. Instead, it is about whether a geometric statement follows logically from some set of axioms.
