median
don steward
mathematics teaching 10 ~ 16

Wednesday, 11 April 2012

creating quadrilaterals

Pierre van Hiele and Michael de Villiers have argued a case for creating (and so defining) special quadrilaterals from transformations
so that side and angle properties are easily deduced as a result of the transformation

[GeoGebra is fine for this]

for a 180 degree rotation of a triangle about the mid-point of a side:



what quadrilateral is generated?
why?

what angle and length properties follow from this transformation (rotation)?



what triangle do you need to start with to generate these quadrilaterals with a half turn about a mid-point:
  • rectangle
  • square
  • rhombus
  • 60, 120 rhombus?  

for a reflection of a triangle:






what shape is created?

what angle and length properties can be deduced from the transformation (reflection)?











using a reflection, how do you create:
  • an arrowhead
  • a rhombus
  • a square?
what special angle property exists when you reflect a right angle triangle (similar to the one above) but not any other triangle?

what angle properties exist when you reflect an isosceles triangle?
an equilateral triangle?

for trapeziums it is not so easy to see how to generate these from a transformation (an isometry)
but you can cheat a bit and use an enlargement
and the 'bit extra' is a trapezium:


what properties of the shape can you deduce from the enlargement transformation?
what extra property is there following an enlargement of an isosceles triangle?

how can a formula for the area of a trapezium be developed from the areas of the two similar triangles when the scale factor is 2 (and in general, using similar triangle ratios)?

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