IMC 2009 Q22
22. A square is divided into eight congruent triangles, as shown.
Two of these triangles are selected at random and shaded black.
What is the probability that the resulting figure has at least one axis of symmetry?
- A \(\dfrac{1}{4}\)
- B \(\dfrac{4}{7}\)
- C \(\dfrac{1}{2}\)
- D \(\dfrac{5}{7}\)
- E 1
Intermediate Mathematical Challenge 2009, Question 22. © UK Mathematics Trust. Original paper on ukmt.org.uk
Answer: D \(\dfrac{5}{7}\)
If the first triangle selected to be shaded is a corner triangle, then the final figure will have at least one axis of symmetry provided that the second triangle selected is one of five triangles. For example, if A is chosen first then there will be at least one axis of symmetry in the final figure if the second triangle selected is B, D, E, G or H. The same applies if an inner triangle is selected first: for example, if B is chosen first then there will be at least one axis of symmetry in the final figure if the second triangle selected is A, C, F, G or H.
So, irrespective of which triangle is selected first, the probability that the final figure has at least one axis of symmetry is \(\dfrac{5}{7}\).
Solution © UK Mathematics Trust, from the IMC 2009 solutions.