"*The definition of a good mathematical problem is the mathematics it generates rather than the problem itself*." — Andrew Wiles from an interview for PBS website on the NOVA program, 'The Proof'.

The following problem is generalized from a question in Mathematics Paper 2 of the 1949 National Senior Examinations for the Union of South Africa.

Given any trapezium *ABCD* with *AB* // *DC* and line *EF* constructed as indicated so that ∠*AEF* = ∠*DCF*. What do you notice about angles *BEC* and *AFD*?

Explore your observation by dragging any of *A*, *B*, *C*, *D* or *E*. Can you deductively explain (prove) why your observation is true?

Is the result still valid if *E* is dragged 'outside' *AD* onto its extension? What if point *D* is dragged towards *C*, and then past *C*? Is the result still valid? Can you deductively explain your observations?

Matric Exam Geometry Problem - 1949

1) Can you formulate alternative, equivalent versions of the result?

2) When, or rather where, is ∠*BEC* a maximum as *E* is dragged along line *AD*? Can you determine the optimal position?

*Hint*: If you get stuck with 2) above, go to this dynamic sketch *"Determining maximum angle"*, or to check your solution.

Check your explorations above, by reading my 2015 paper in *At Right Angles* and *Learning and Teaching Mathematics* at *"Flashback to the Past: a 1949 Matric Geometry Question"*.

*Back
to "Dynamic Geometry Sketches"*

*Back
to "Student Explorations"*

Created by Michael de Villiers, 13 August 2015.