The gradient of the blue triangle and the gradient of the red triangle are not the same. So the hypotenuse of the top triangle is not a straight line. If you were to join the two points together (top point of blue triangle with bottom point of red triangle), you would notice a thin area between that line and the two triangles. The area of this would be 1, so that when you rearrange the triangles it looks like you have gained a square.

Gradient of red triangle is 3/8, gradient of blue triangle is 2/5, gradient of whole triangle is 5/13

This is impossible unless the area is very slightly changed. It seems like the red triangles differ in that one seems to have slightly bigger borders. The red triangle would by default be the cultprit because it is the biggest piece and modifying its area without the reader noticing is easier. Once this is established, the rest of the puzzle is simply moving around the rest of the pieces. The gold piece does not create a perfect fit, so it has a hole but other than that, the whole shape is a triangle.

...This has me scratching my head. There's something not right, right from the start. The original triangle has an area of 32.5 square units. (0.5 * 13 * 5 = 32.5). The areas of the individual pieces add up to 32 square units.

Red triangle = 0.5 * 8 * 3 = 12

Green triangle = 0.5 * 5 * 2 = 5

Orange piece = 7

Green piece = 8

12 + 5 + 7 + 8 = 32

(I should have read Mohawk20 answer first. She's got it)