Purpose of This
Ever since ancient times when the Greek philosopher and mathematician Pythagoras of Samos (c. 570–495 BC) spent his afternoons drawing right-angled triangles in the sand with a stick, mathematicians have been flummoxed by the “Pythagorean Paradox” ... that is, the anomaly of a right-angled triangle’s “hypotenuse” (the longest side of a right-angled triangle) being “parallel” with one of the triangle’s legs—as pertains to the particular circumstance of the right-angled triangle that is used in the construction of an “Ellipse,” or in the construction of a “Hyperbola.”
This web page will attempt to demystify the “Pythagorean Paradox” through the use of several graphing illustrations.
We start with a “horizontal” Ellipse (the light-blue oval above) which has a “vertex” of length five, measured from the center of the Ellipse. The center of this particular Ellipse is also the center of the graph that the Ellipse has been constructed on.
The “focus” of the Ellipse is of length four, also measured from the center of the Ellipse.
Horizontal Hyperbola, with Asymptotes and Rectangular Box
Next, we have a “horizontal” Hyperbola (with its two, maroon-colored “branches” displayed above) with a “vertex” of length four, measured from the center of the Hyperbola. The center of this particular Hyperbola is also the center of the graph that the Hyperbola has been constructed on.
The “focus” of the Hyperbola is of length five, measured from the center of the Hyperbola. You have probably already noticed that the positions of the vertex and focus on the graph are reversed from their positions on the graph of the Ellipse.
Ellipse, Hyperbola, Asymptotes, and Circle
And finally, we have both the Ellipse and Hyperbola on the same graph. Also added to the graph is a Circle (with “radius” five) that will come in handy in a second when we add a right-angled triangle to the graph.
Ellipse, Hyperbola, Asymptotes, Circle, Right Triangles and Rectangular Box
(Two Togglable Images)
[or 'tab-to,' then press the 'enter' key]
to toggle between two graphing images,
along with their accompanying text underneath.
Screenshot below of an adapted
desmos.com calculator graph