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Completing Quadrilaterals

Age 11 to 14
Challenge Level Yellow star
Secondary curriculum
  • Problem
  • Getting Started
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Completing Quadrilaterals printable sheet


Each line is one side of the named quadrilateral. Can you draw the other three sides in the 4 by 4 dotty grids?

If there is more than one possibility, try to find the quadrilateral with the largest area (given in brackets).

An * indicates that there is a “special” quadrilateral with a larger area than the one
given in the brackets. For example, the largest parallelogram might be a square or a rectangle. Take a look at the Getting Started page if you want to find out more about special cases.

   1.  Rectangle (6)
 
   2.  Square (8)
 
   3.  Rectangle (4)
 


   4.  Isosceles
       Trapezium (12)
 

   5.  Parallelogram (9*)


   6.  Kite (8)
 


   7.  Parallelogram (6)
 


   8.  Square (5)
 


   9.  Kite (12)
 


   10.  Rhombus (4*)
 

   11.  Parallelogram (3*)

   12.  Kite (6)


   13.  Arrowhead (6)
        (Concave Kite)
 
   14.  Kite (8)    15.  Rhombus (8)

   16.  Rhombus (3)
 


   17. Arrowhead (4)
      (Concave Kite)

   18.  Trapezium (9)

   19.  Parallelogram (8)
 


   20.  Isosceles
        Trapezium (8)
 
   21.  Kite (3)


   22.  Arrowhead (4)
        (Concave Kite)
 
   23.  Kite (9)    24.  Trapezium (9)



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

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Linkage

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?

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.

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The NRICH Project aims to enrich the mathematical experiences of all learners. To support this aim, members of the NRICH team work in a wide range of capacities, including providing professional development for teachers wishing to embed rich mathematical tasks into everyday classroom practice.

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