don steward
mathematics teaching 10 ~ 16

Friday, 6 April 2012

one less side, same area

Paul Andrews (was at Cambridge University now at Stockholm, Sweden) has observed that in several European countries they present the problem of creating a triangle equal in area to a quadrilateral:

with one straight line, create a triangle equal in area to the quadrilateral (blobs are at the 4 corners)

students often respond by drawing a line that bisects the two orange sides - but does this work?

once found, involving parallel lines, the beauty of this transformation (as Paul often asserts) is that any shape can be reduced, stepwise, to a triangle

as every triangle is equal in area to a rectangle
then as far as area is concerned, every shape (with straight line edges) is a rectangle in disguise...

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