Geometry problem, borrowed and extended. Try it !

I borrowed this from http://fivetriangles.blogspot.com/
184. *** Overlapping sectors

In the diagram is rectangle ABCD with height 10 cm. An arc with centre at point B is drawn from point A to side BC. An arc with centre at point C is drawn from point D to side BC. Given that the shaded (coloured) regions ⓐ and ⓑ have equal area, determine the length of BC.

The extension is “and where do the quarter circles cross?”
It could be a two line calculation !

-1 x -1 = 1, but some need convincing

Here is a popular argument:   -1 x -1 has to be 1 or -1

If it was equal to -1 then -1 x -1 = -1 x -1 x 1 = -1 x1 and so dividing both sides by -1 we get -1 = 1, which is not a good idea!, hence -1 x -1 = 1

This argument begs so many questions that it is difficult to know where to start.

Here is a much better one, but it does stretch the idea of area a little :

From the diagram  (a – 1) x (b – 1) = a x b – a – b + 1

Set a = 0 and b = 0 to get  (0 – 1) x ( 0 – 1) = 1, and since 0 – 1 is equal to -1        we get -1 x -1 = 1

This has some connection with evaluating for example  3 x ( 8 – 2)  using the distributive law.

The distributive law is a law for  a(b + c) and says nothing about  a(b – c), but never mind, we go gaily about the common task.