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It's a square with vertices at (5,0), (0,-5), (-5,0) and (0,5).

So the area = 4 * triangles (one of them : (0,0), (5,0), (0,5)) = 5*5/2 * 4 = 50 :)

Fig, I don't follow. Please explain the bolded portion.

To know the area, I build up it by using the area of 4 similar triangles .... Then, I calculate the area of one of them. This area is also the area of the 3 other triangles.

Have a look on the Fig 1, the triangle in flashy green is the one described in my calculation of the area.

I got ya'. but could we do the same with 1,4; -1,-4; -1,4; 1,-4 or 3,2; -3,-2; -3,2; 3,-2. I tried them cursorily. they didn't seem to work.

The 2 shapes u design like this are rectangles. ... Notice the x and y coordonates of the point u have built up : their absolute value is not equal (|1| != |4|) .... If they were equal, then we have a square

Notice also that sides of such rectangle are parallel to the X and Y axes

If gives a and b the absolute values of the x and y coordonaites of each vertex from a same rectangle, then we have:

The aera of such a rectangle = (2*a) * (2*b)

In your cases,
o Aera 1 = 2*1 * 2*4 = 16
o Aera 2 = 3*1 * 2*2 = 12

The 2 shapes u design like this are rectangles. ... Notice the x and y coordonates of the point u have built up : their absolute value is not equal (|1| != |4|

In the stem, we need the abs. val. of x and y to equal 5. Why would we need the abs. val. of x and y (|1| and |4| in this case) to be equal. Don't they just need to sum to 5?

The 2 shapes u design like this are rectangles. ... Notice the x and y coordonates of the point u have built up : their absolute value is not equal (|1| != |4|

In the stem, we need the abs. val. of x and y to equal 5. Why would we need the abs. val. of x and y (|1| and |4| in this case) to be equal. Don't they just need to sum to 5?

No .... It is 2 different things.... Your example are not equivalent to the problem

|x| + |y| = b (b>0) will always create a square such that it is shown on my Fig 1.

Your examples create rectangles with sides that are parellel to the X and Y axes .... Draw them, u will see what I mean

An equivalent system of equations for your examples is :
o |x| = c
o |y| = d