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Given a square ABCD of sides 10 cm, and using the corners as centres, construct four quadrants with radius 10 cm each inside the square. The four arcs intersect at P, Q, R and S. Find the area enclosed by PQRS.

### Darts and Kites

A rhombus PQRS has an angle of 72 degrees. OQ = OR = OS = 1 unit. Find all the angles, show that POR is a straight line and that the side of the rhombus is equal to the Golden Ratio.

### Hexy-metry

A hexagon, with sides alternately a and b units in length, is inscribed in a circle. How big is the radius of the circle?

##### Stage: 4 Challenge Level:

Why do this problem?
This problem involves the interpretation of a very simple concrete structure, a linkage of 4 rods, and the angles that the quadrilateral formed by the rods could make if the joints between the rods at the vertices are totally flexible. Experimental evidence will offer ideas which then need justification and proof by forming convincing arguments.

The solution uses the cosine and sine rules. To find the constraints on the angles in the general case requires an argument using inequalities.

Possible approach
You might allow time for learners to explore the quadrilateral using strips of card or plastic and split pins, or a dynamic geometry package. This will help them to identify what can be varied and what not.

Discuss the freedoms and constraints within the problem, the impact these might have and how they could influence the structure of any investigation (what can be changed and what cannot).

Encourage groups to identify ideas that they would like to investigate. Spend time planning what they might do and sharing ideas before developing them.

Share findings and approaches.

Key questions