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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?

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

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The Cyclic Quadrilateral

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.

Completing Quadrilaterals

Stage: 3 Challenge Level: Challenge Level:1

Join dots on each grid below to make the named quadrilateral.
You must use the side given, you can't shorten or extend it.

If there is more than one possibility, try to find the one with the largest area.

You will need to make decisions about whether you want to allow special cases in your solutions. For example, will you allow a square if a rectangle is asked for? Take a look at the Getting Started page if you want to find out more about special cases.


1  Rectangle
 
2  Square
 
3  Rectangle
 

4  Isosceles Trapezium
 

5  Parallelogram
 

6  Kite
 

7  Parallelogram
 

8  Square
 

9  Kite
 

10  Rhombus
 
11  Parallelogram 12  Kite

13  Arrowhead
 
14  Kite 15  Rhombus

16  Rhombus
 
17  Arrowhead 18  Trapezium

19  Parallelogram
 
20  Isosceles Trapezium 21  Kite

22  Arrowhead
 
23  Kite 24  Trapezium


You may find it useful to print this worksheet of the problem.

If you want to check you have the quadrilaterals with the largest area, take a look below:


1 rectangle:  6 
2 square:  8 
3 rectangle:  4

4 isosceles trapezium:  12
5 parallelogram:  10  (9 if you don't allow squares)
6 kite:  8

7 parallelogram:  6
8 square:  5
9 kite:  12

10 rhombus:  5 (4 if you don't allow squares)
11 parallelogram:  6 (3 if you don't allow rectangles)
12 kite:  6

13 arrowhead:  6
14 kite:  8
15 rhombus:  8

16 rhombus:  3
17 arrowhead:  4
18 trapezium:  9

19 parallelogram:  8
20 isosceles trapezium:  8
21 kite:  3

22 arrowhead:  4
23 kite:  9
24 trapezium:  9


With thanks to Don Steward, whose ideas formed the basis of this problem.