median
don steward
mathematics teaching 10 ~ 16

## Monday, 28 March 2016

### right trapeziums

the Ancient Babylonians (up to 4000 years ago) appear to have worked substantially with right-angled triangles and right angled trapeziums

one of their many clever techniques is that of being able to split a right-angled trapezium into two equal areas (seemingly for inheritance purposes - with fields and orchards)

in a study reported early in 2016, Mathieu Ossendrijver claimed to show that the Babylonians had tracked the path of Jupiter

one of the techniques involved in this calculation involved splitting a right-trapezium into two equal areas

splitting a right trapezium into two equal areas is a difficult problem
we considered the simpler problem of finding the length of a line part way along a right-trapezium

several techniques can be used to find this length

using steps: the Babylonians had an understanding of what gets translated as 'feed': how much you go down (or up) for a certain distance across
(i.e. the tan of the angle of depression)
or you can look at the areas of trapeziums

or straight line graph equations

or similar triangles

in some of these questions the line does split the area in two

which?

for a base split in the ratio 1 : 2
how do you find the length 'n'?

how do you do this in general?

turning to the much harder problem of how to dissect the area

using the result previously obtained for the distance 'n' being the weighted mean of 'a' and 'b' in the ratio 1 : 2

using the areas of the trapeziums
left = right

(or you could also use e.g. left = half of the whole)