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Question Number 116770 by bounhome last updated on 06/Oct/20

h o w t o p r o v e t h e P y t h a g o r e a n t h e o r e m (c^{2} = a^{2} + b^{2})

Commented by $@y@m last updated on 06/Oct/20

https://www.math-only-math.com/proof-of-pythagorean-theorem.html#:~:text=The%20proof%20of%20Pythagorean%20Theorem,of%20c%20(c2).

Answered by MJS_new last updated on 06/Oct/20

α + β + γ = 180 ° \land γ = 90 ° \Rightarrow β = 90 ° - α

\frac{a}{\sin α} = \frac{b}{\sin β} = \frac{c}{\sin γ}

\frac{a}{\sin α} = \frac{b}{\sin (90 ° - α)} = \frac{c}{\sin 90 °}

\frac{a}{\sin α} = \frac{b}{\cos α} = c

\Rightarrow a = c \sin α \land b = c \cos α

a^{2} + b^{2} = c^{2} \sin^{2} α + c^{2} \cos^{2} α =

= c^{2} (\sin^{2} α + \cos^{2} α) = c^{2}

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