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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:

Suppose one angle is 60 degrees as shown in the diagram.

Suppose the angle $s$ is $60$ degrees, then it is easy to calculate the length of the diagonal and from that to calculatethe opposite angle in the diagram.

You might like to check your answer by drawing the quadrilateral accurately, using ruler and compasses only, and then measuring the angles.

Calculate the other angles of the quadrilateral.

Now calculate the angles of the cyclic quadrilateral formed by keeping the lengths of the sides the same and changing the angles so that opposite angles add up to 180 degrees. You might wish to use a spreadsheet.

 The question states that the quadrilateral is convex; this means that the angles $s$ and $q$ are at most $180$ degrees. Imagine moving the rods to make the angle $s$ as large or as small as possible. Find the largest and smallest values of $s$ and $q$.

To calculate the angles of the cyclic quadrilateral formed by keeping the lengths of the sides the same and changing the angles so that opposite angles add up to $180$ degrees you simply need to use the fact that, in this case, $\cos s = - \cos q$.