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.

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By Michael de Villiers. Created with *WebSketchpad*, 26 June 2022.