The "successful" region has half the size of the whole triangle, so by similar triangles, the probability that the area of triangle XYD is at most 12 is 1/4.
If you draw the triangle out on a grid papar and draw a line across the 2 marker point from XY up of the YZ side, you will see that the line from that marker point across to the hypotenuse is actually 8. Then you find the area of the little triangle with legs 8 and 4 (since 6-2 is 4), and the area is 16. After that, find the total area of the big triangle, which is 12x6/2=36. Since we want to know the area of the quadrilateral at the bottom of the big triangle seperated by the 2 marker point, we subtract 16 from 36, which equals to 20. 20/36=5/9. And that is my final answer. I tested it and it is correct. I hope this helps.
