This study aimed at estimating the relationships between linear type traits and milk production in the dual-purpose Aosta Red Pied (ARP) cattle breed, by expressing type traits as factor scores with the same biological meaning of the individual traits. Factor analysis was applied to individual type traits for muscularity and udder of 32,275 first-parity ARP cows, obtaining 3 factor scores for individual muscularity (F1), udder side (F2), and udder conformation (F3). Data from 169,008 test-day records of milk, fat, and protein yield (kg), belonging to the first 3 lactations of 16,605 cows, were also analyzed. After obtaining genetic parameters for both morphological factors and milk production traits through a series of AIREML single-trait models, bivariate analyses were performed on a data set accounting for 201,283 records of 35,530 cows, to assess the phenotypic and genetic correlations among all factor scores and milk yield traits. The heritability estimates obtained proved to be moderate for both groups of traits, ranging from 0.132 (fat) to 0.314 (F1). Muscularity factor showed moderate and negative genetic correlations (ra) with udder size (-0.376) and udder conformation (0.214) factors. A low and negative ra was found between udder factors. Strong and positive ra were found among all the 3 milk production traits and F 0010 (ra≥0.597). Negative ra with milk traits were obtained for both F 0005 and F3, ranging from -0.417 to -0.221. Phenotypic correlations were lower than the genetic ones, and sometimes close to zero. The antagonism between milk production and meat attitude traits suggests that great attention should be paid in assigning proper weight to the traits, comprising functional traits such as udder conformation, included in selection indices for the dual-purpose breed. The ra obtained for factor scores are consistent with previous estimates for the corresponding individual type traits, and this confirms the possible use of factor analysis to improve type traits relevant to beef attitude.