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Once students come to the conclusion that all of their examples have four congruent triangles, invite them to come up with convincing arguments leading to a proof that it is impossible for the triangles to have the same perimeter unless they are congruent.
Can you now add a third triangle with the same perimeter?
And a fourth?
Building Tetrahedra
Age 14 to 16
Challenge Level 
Why do this problem?
This problem invites students to explore the properties of 3D shapes, and in particular, tetrahedra.Possible approach
The challenge is for students to create a non-regular tetrahedron where all four of the triangular faces have the same perimeter (you could specify a particular perimeter such as 20cm). Students could work with nets and sketches or alternatively create 3D models of tetrahedra in order to explore the constraints on the triangles that make up a tetrahedron, and how that is affected by the additional constraint that the triangles have the same perimeter.Once students come to the conclusion that all of their examples have four congruent triangles, invite them to come up with convincing arguments leading to a proof that it is impossible for the triangles to have the same perimeter unless they are congruent.
Key questions
Imagine putting together two triangles with a perimeter of 20, edge to edge.Can you now add a third triangle with the same perimeter?
And a fourth?
Possible extension
Tet-TroublePossible support
Constructing Triangles invites students to think about similar constraints in a 2D context.You may also like
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