IMC 2012 Q25
25. The diagram shows a ceramic design by the Catalan architect Antoni Gaudi. It is formed by drawing eight lines connecting points which divide the edges of the outer regular octagon into three equal parts, as shown.
What fraction of the octagon is shaded?
- A \(\dfrac{1}{5}\)
- B \(\dfrac{2}{9}\)
- C \(\dfrac{1}{4}\)
- D \(\dfrac{3}{10}\)
- E \(\dfrac{5}{16}\)
Intermediate Mathematical Challenge 2012, Question 25. © UK Mathematics Trust. Original paper on ukmt.org.uk
Answer: B \(\dfrac{2}{9}\)
We consider the triangular segment of the octagon formed by joining two adjacent vertices, \(P\) and \(Q\) to the centre, \(O\). For convenience, we show this segment, drawn on a larger scale, on the left, where we have added the lines \(RW\), \(ST\), \(TW\) and \(UV\). These lines are parallel to the edges of the triangle \(POQ\), as shown and together with the lines \(RU\) and \(SV\) they divide the triangle \(OPQ\) into 9 congruent triangles, of which 2 are shaded. Thus \(\dfrac{2}{9}\) of the segment is shaded. The same holds for all the other congruent segments of the octagon. So \(\dfrac{2}{9}\) of the whole octagon is shaded.
Solution © UK Mathematics Trust, from the IMC 2012 solutions. Longer versions with investigations: Solutions and investigations (PDF).