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

aeα+be~α+cex=0a{\bold e}_\alpha + b\tilde{\bold e}_\alpha + c{\bold e}_x = \bold 0(1)

with

eα=(cos⁡αsin⁡α)ande~α=(−sin⁡αcos⁡α){\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 cexc{\bold e}_x in the loop closure equation (1)

−cex=aeα+be~α-c{\bold e}_x = a{\bold e}_\alpha + b\tilde{\bold e}_\alpha

and squaring

finally results in the Pythagorean theorem (2)

c2=a2+b2c^2 = a^2 + b^2(2)

More Triangle Stuff

Introducing the hypothenuse segments p=a⋅exp={\bold a}\cdot{\bold e}_x and q=b⋅exq={\bold b}\cdot{\bold e}_x, we can further obtain the following useful formulas.

segment p segment q height h area
cp=a2cp = a^2 cq=b2cq = b^2 pq=h2pq = h^2 ab=chab = ch