An Area Preserving Transformation: Shearing

Shearing is a special affine transformation that preserves area, and is often useful in geometric problem solving and proving of theorems. Probably the most well-known example at high school level of a straightforward application of shearing is the theorem that two triangles with the same base and same height have the same area.

Below are some introductory examples of shearing, first in coordinate form, and then an example is given for an area ratio problem with regard to a parallelogram, and finally two simple proofs of the theorem of Pythagoras by using shearing. Use the Link buttons in the sketch to navigate to the relevant dynamic sketch.

An Area Preserving Transformation: Shearing

Read up further about 'shearing' and other affine transformations like 'stretching', and their applications, and write a report.

De Villiers, M. (1993). Transformations: A golden thread in school mathematics. Spectrum, 31(4), 11-18.
De Villiers, M. (2003). The Affine Equivalence of Cubic Polynomials. KZN Mathematics Journal, November, 5-10.
De Villiers, M. (2004). All cubic polynomials are point symmetric. Learning & Teaching Mathematics, No. 1, April, 12-15.

Related Links
Miscellaneous Dynamic Transformations of Geometric Figures & Graphs
International Mathematical Talent Search (IMTS) Problem Generalized
Sylvie's Theorem
Some Parallelo-hexagon Area Ratios
Area Parallelogram Partition Theorem: Another Example of the Discovery Function of Proof

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Created by Michael de Villiers, 3 June 2022 with WebSketchpad; updated 4 Oct 2023 with references & related links.