Let A=[abcd].
How could we show that ad−bc is the area of a parallelogram with vertices (0,0), (a,b), (c,d), (a+b,c+d)?
Are the areas of the following parallelograms the same?
(1) parallelogram with vertices (0,0), (a,b), (c,d), (a+c,b+d).
(2) parallelogram with vertices (0,0), (a,c), (b,d), (a+b,c+d).
(3) parallelogram with vertices (0,0), (a,b), (c,d), (a+d,b+c).
(4) parallelogram with vertices (0,0), (a,c), (b,d), (a+d,b+c).
Thank you very much.
Answer
Spend a little time with this figure due to Solomon W. Golomb and enlightenment is not far off:
(Appeared in Mathematics Magazine, March 1985.)
Attribution
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