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Right Triangles: Instructor's Guide

Main Ideas

Hyperbola geometry is similar to circle geometry — but different.

Students' Task

Estimated Time: 10 minutes, including wrap-up

Students should be asked to answer the following questions on their whiteboards:
Draw a 3–4–5 triangle.

Some students will presumably ask, “Euclidean or hyperbolic?”, at which point the question should be changed to:
Draw a 3–4–5 triangle in Euclidean geometry.
Then draw a 3–4–5 triangle in hyperbola geometry.

Prerequisite Knowledge

Basics of hyperbola geometry.



Point out that the scaling in hyperbola geometry is different in different directions, but nonetheless linear in any given direction — and in fact equal in orthogonal directions, such as along the axes.

Point out that “angles” only exist in hyperbola geometry between two spacelike directions, or between two timelike directions. “Right angles” are not angles! Right triangles only have one angle!

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