Radioactively labelled amino acids will be found in the ribosomes. These are the organelles that are the site of protein synthesis.
Amino acids are taken up into the cytoplasm from the surrounding cell culture medium. Amino acids are then bound to tRNAs (with the enzyme aminoacyl-tRNA synthetases) and taken to the ribosome, where they are assembled into a polypeptide chain by the translation machinery.
