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Four rods, two of length a and two of length b, are linked to form a kite. The linkage is moveable so that the angles change. What is the maximum area of the kite?

### Making Rectangles, Making Squares

How many differently shaped rectangles can you build using these equilateral and isosceles triangles? Can you make a square?

This gives a short summary of the properties and theorems of cyclic quadrilaterals and links to some practical examples to be found elsewhere on the site.

# Property Chart

### Why do this problem?

This game provides an interesting context in which to consider the properties of quadrilaterals (or triangles), and has a particular focus on the combinations of properties that are possible.

### Possible approach

Quadrilaterals Game could be used either at the beginning or at the end of this problem.

Use the instruction on the problem page to set up and play the game for about half a lesson, then move the group on to the questions at the end of the problem.

### Key questions

• Which shapes are most useful in this game?
• Which property cards are 'good' and 'bad' and why?
• Tell me two cards where there is no shape that works for both.

### Possible support

The game could be played as a whole class - shuffle and arrange the property cards on the board so that everyone has the same question. Groups of 3 or 4 then work together filling in the grid and checking each others work. A correct shape or gap will earn 10 points, but each incorrect shape or gap will lose 10 points. After a set time, all the grids are displayed, and students try to find errors in the other groups' work, in order to establish the scores and the winners. They may be ready to try the problem as stated after this!

Another way in to the problem could be to produce some partly completed grids and ask students to finish them, or produce some completed grids with a few deliberate errors for students to find and correct.

### Possible extension

If only the quadrilaterals are visible on the board can you identify the property cards in each position? In what other ways can you adapt/invert/develop this game to make new and possibly harder challenges?

Suggest students have a go at Shapely Pairs