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Pericut

Two semicircle sit on the diameter of a semicircle centre O of twice their radius. Lines through O divide the perimeter into two parts. What can you say about the lengths of these two parts?

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Tri-split

A point P is selected anywhere inside an equilateral triangle. What can you say about the sum of the perpendicular distances from P to the sides of the triangle? Can you prove your conjecture?

Triangles Within Pentagons

Stage: 4 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

Having identified the rule is it possible to work backwards and reconstruct pentagons using the triangular numbers?

This problem builds on the work of the three problems "sequences and series ", "triangles within triangles " and "triangles within squares ".

There are many more patterns and relationships involving triangular numbers and some experimentation could result in some pupil generated discoveries.