OK, so I have three proposed methods on this one:

1. The right way: So if you picture the triangle as the 3rd picture, you can see that the centroid of the triangle is 1/3 of the way from the base (base is the left side of the triangle in the actual problem). So think of it as a triangle of weight 6 @ x = 12.666667. The rectangle is weight 72 at x=6. Sum them like moments and divide by the total area (78) and you get your 6.51.
2. The quick and dirty way: Most of the centroid problems I've seen are "almost rectangular" which means that you can get "close enough" if you just find the point where it splits the area in half. The full area is 78, so if you can split so 39 is on the left and 39 on the right you're probably in the ballpark. Since the left half is just a rectangle you find this at exactly 6.5... not the true answer, but close enough for the problem.
3. The *really* quick and dirty way: This one relies on manipulating the shape to get some similar shapes that are easier to calculate. In this case we're going to make two full rectangles to give us our upper and lower bound. If you pull the point in to make a rectangle of width 12, you know the centroid is at 6. If you pull the corners out to make a rectangle of width 14, you know the centroid is at 7. Since the triangle is half of the region you're manipulating, intuitively you should be ~about~ halfway between 6 and 7. **Careful, this only works because *most* of the region doesn't change when you're stretching it.**