one of the dish rack tines broke when I forced a large platter into the rack

It took a few minutes for me to reallize that I needed to look at the way/direction the tine clip was in the existing position. It is important to mesh the part into all the correct positons before trying to complete the repair into the final snap/placement of the tine/clip. A minute or two of looking and (really important) not trying to replace both front AND rear tines at the same time! They are mirror images of each other. If you change one at a time, you can use the other as a template!