# Right Triangles

## Prompt

“Draw a 3–4–5 triangle.”

Some students will presumably ask, “Euclidean or hyperbolic?”,
at which point the prompt should be changed to:

“Draw a 3–4–5 triangle in Euclidean geometry.”
“Then draw a 3–4–5 triangle in hyperbola geometry.”

## Context

Estimated Time: 10 minutes, including wrap-up

This SWBQ develops understanding of hyperbola geometry by contrasting with known results in circle geometry, thus emphasizing that Hyperbola geometry is similar to circle geometry — but different.

Students should be familiar with the basics of hyperbola geometry.

## Wrap Up

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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