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The Right Triangle

Base Geometry

Let the right triangle hypothenuse be aligned with the coordinate system x-axis. The vector loop closure equation running counter-clockwise then reads

$$a{\bold e}\alpha + b\tilde{\bold e}\alpha + c{\bold e}_x = \bold 0$$ (1)

with

$${\bold e}\alpha = \begin{pmatrix}\cos\alpha\ \sin\alpha\end{pmatrix} \quad and \quad \tilde{\bold e}\alpha = \begin{pmatrix}-\sin\alpha\ \cos\alpha\end{pmatrix}$$

Resolving for the hypothenuse part c \bold e x in the loop closure equation (1)

$$-c{\bold e}x = a{\bold e}\alpha + b\tilde{\bold e}_\alpha$$

and squaring

finally results in the Pythagorean theorem (2)

c 2 = a 2 + b 2 (2)

More Triangle Stuff

Introducing the hypothenuse segments p = \bold a \bold e x and q = \bold b \bold e x , we can further obtain the following useful formulas.

segment p segment q height h area
c p = a 2 c q = b 2 p q = h 2 a b = c h