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Four rods are hinged at their ends to form a convex quadrilateral. Investigate the different shapes that the quadrilateral can take. Be patient this problem may be slow to load.

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

### Why do this problem?

This problem requires the solver to reason geometrically and make use of symmetry. By re-presenting the information in a different way, for example by adding additional lines (a useful technique in geometrical problems) more structure can be revealed. It is an interesting idea that adding something, and therefore apparently making it more complex, can sometimes make a problem more accessible. Then of course there is an opportunity to use the cosine rule in a non-standard context.

### Possible approach

Use the images in this document to make cards.

Display the problem and ask learners to work in groups rearranging the cards in ways which help to make connections for them.

Share ideas and relationships that groups notice before going on to solve the problem. Encourage careful reasoning and convincing arguments. For example:

• "Why can you say those two angles are equal?

There are several ways of solving the problem so share different approaches and discuss what helped to move thinking forward.

Two observations may be worth drawing attention to :

• the reflection symmetry between $a$ followed by $b$ and $b$ followed by $a$,
• the angle at the circumference as half the angle subtended at the centre onto the same arc (in this case an arc greater than a semicircle) .

### Key questions

• How can you use the symmetry of the figure to determine some of the angles ?
• What things in the images helped to make connections for you?
• What do we need to find, and can that be found directly ?

### Possible extension

For a similar level challenge try Bendy Quad

And for something much harder try Biggest Bendy

### Possible support

Concentrate on the interesting property that all angles in the hexagon are equal. This might be approached practically: drawing a circle, split into thirds to emphasise the symmetry, then splitting each third into the same two unequal proportions before measuring all six angles.

Ask the students if they can explain why these should all be equal .