Doesgeometryhavea placein art and design?Mostpeoplewouldguessthat they do. We’ll get a chance
to start to see how whenwe examinetessellationsand symmetryin futurelessons.
LessonExercises
- Explainwhy the compositionof two or moreisometriesmustalso be an isometry.
- Recallthe glidereflectionin example1. Supposethis glidereflectionis appliedto a triangle,and then
appliedagainto the imageof the triangle.Describehow the final imagecomparesto the originaltriangle. - Whatone basictransformationis equivalentto a reflectionin two parallellines?
- A pointis reflectedin line. The imageis reflectedin line. , and the two linesare units
apart.Whatis the distancefrom the originalpointto the final image? - Point is reflectedin two parallellines.Doesit matterin whichline is reflectedfirst?Explain.
- Whatone basictransformationis equivalentto a reflectionin two perpendicularlines?
- Point is reflectedin the two axes.Doesit matterin whichaxis is reflectedfirst?Explain.
- Prove:Reflectionin , followedby reflectionin , is equivalentto a rotation.
- The glidereflectionin example1 is appliedto the “donut”below. It is reflectedover the - axis, and the
imageis then translated unitsto the right.The sameglidereflectionis appliedagainto the imageof the
first glidereflection.Whatare the coordinatesof the centerof the donutin the final image?
- Describehow the final imageis relatedto the originaldonut.
Answers
- Imagesin the first isometryare congruentto the originalfigure.The sameis true of the secondisometry.
If is a polygon, the imageafter the first isometry, and the imageof after the secondisometry,
we knowthat and So i.e., the final imageis congruentto the original.
- The originaltrianglehas moved unitsto the right.
- A translation