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Which variables in the problem are rational and which are irrational?
Can we make use of the angle properties of the triangles?
Impossible Triangles?
Age 16 to 18
Challenge Level 
Why do this problem
This problem will make students experiment, conjecture and prove. They will need to understand the interplay between rational and irrational numbers.Possible Approach
This activity lends itself to hands-on experimentation with
cut-outs of triangles before algebraic proof is attempted. Without
cut-out shapes it is a good exercise in visualistion. Students
might need to be encouraged to pursue various lines of
enquiry.
Explaining a proof clearly to someone else is a very good way
to discover where the holes in the argument are. You could ask
students to explain their proofs to each other in pairs and then to
the whole class. Can the class point out areas where proofs or
explanations are unclear?
Key Questions
How do we relate side-length and area of triangles?Which variables in the problem are rational and which are irrational?
Can we make use of the angle properties of the triangles?
Possible Extensions
Can you suggest other regular shapes which you could prove to be impossible?Possible Support
Try the easier question Equal
Equilateral Triangles first.
Struggling students could be asked to hear other students
proofs and comment on their clarity.
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