The following problem was used as Question 25 in the Senior (Grades 10-12) Round 2 of the South African Mathematics Olympiad in 2022.
Question 25
A square with side length AB = 10 cm is placed as shown below over two parallel lines, which are a distance of 10 cm apart.
Determine the sum of the perimeters of triangles AEF and GCH.
SA Mathematics Olympiad 2022, Round 2, Q25
Investigate
Drag any of the vertices A, B, C or C. What do you notice about the sum of the perimeters of the two 'overlapping' triangles?
Use your observation to place the square in a more convenient position, and use it to show that the sum of the perimeters is equal to 20 cm.
Challenge
Can you explain why (prove that) the sum of the two perimeters remains constant, irrespective of how the square is placed over the two parallel lines?
Further Investigation
1) Drag either one of the two integer sliders for AB and h so that AB ≠ h. Next drag again any of the vertices A, B, C or D to move the square.
What do you now notice about the sum of the perimeters of the two 'overlapping' triangles?
2) Again drag either one of the two integer sliders for AB and h until AB = h (but not equal to 10). Next drag again any of the vertices A, B, C or D to change the position of the square.
What do you now notice about the sum of the perimeters of the two 'overlapping' triangles?
3) Formulate a general conjecture in terms of AB and h, and prove it.
Note
A copy of SAMO R2 2022 is available at question paper, with the worked solutions available at answers.
Acknowledgement
I'm grateful to Erik Bilsted, former mathematics lecturer at UC Lillebælt, Denmark & Høgskolen Bergen, Norway, and currently honorary lecturer, University of KwaZulu-Natal, for bringing this interesting problem to my attention.
Back
to "Dynamic Geometry Sketches"
Back
to "Student Explorations"
By Michael de Villiers. Created with WebSketchpad, 26 June 2022.