Let's begin by sketching the rectangle described in the question stem:

Now, if the rectangle is to be cut into exactly two squares, we have only one possible scenario - we can divide the length of the rectangle into two segments of 10, each of which forms a side of one square. The width remains undivided with a length of 10; this dimension also serves as a side of each square. We are left with two squares with dimensions of 10 x 10.

Note that we cannot divide the width of the rectangle into two segments to form the two squares. Each segment would be less than 10 - too short to form a square with the length of the rectangle.

Now we must divide each square into two triangles. The only way to accomplish this is to divide the square along its diagonal to form two equal 45:45:90 triangles. Any other division would result in other shapes, as shown below:

Since each square is composed of two triangles of equal size, the area of one triangle must equal half the area of the square. The area of the square is 10 x 10 = 100, so the area of the triangle must be 50.