PlanetMath: proof of Pythagorean theorem

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proof of Pythagorean theorem

(Proof)

Let be a right triangle with hypotenuse . Draw the height .

$\includegraphics{pythaproof}$

Using the right angles $\angle BAC$ and $\angle ATB$ and the fact that the sum of angles on any triangle is $180^\circ$ , it can be shown that

$\begin{eqnarray*} \angle BAT &=& \angle ACT\ \angle TAC &=& \angle CBA \end{eqnarray*}$

and therefore we have the following triangle similarities:

$\begin{displaymath}\triangle ABC \sim \triangle TBA \sim \triangle TAC.\end{displaymath}$

From those similarities, we have $\frac{AB}{BC}=\frac{TB}{BA}$ and thus $AB^2 = BC\cdot TB$ . Also $\frac{AC}{BC}=\frac{TC}{AC}$ and thus $AC^2= BC \cdot TC$ . We have then

$\begin{displaymath}AB^2 + AC^2 = BC(BT+TC) = BC\cdot BC = BC^2\end{displaymath}$

which concludes the proof.

"proof of Pythagorean theorem" is owned by drini. [ owner history (1) ]

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Cross-references: proof, similarities, triangle, angles, sum, right angles, height, hypotenuse, right triangle
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This is version 5 of proof of Pythagorean theorem, born on 2002-06-23, modified 2002-06-25.
Object id is 3130, canonical name is ProofOfPythagoreanTheorem.
Accessed 5249 times total.

Classification:

AMS MSC:

51-00 (Geometry :: General reference works )

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