Hi,
Brazilian test is an indirect experimental method to measure the tensile strength of rock.
Because the strength of rocks is more pronounced in compression and shear, the tensile strength is less significant. However, tensile strength (along with the triaxial compression strength) can be used to obtain precisly the Mohr criterion of failure (cohesion and compression).
The tensile strength is also required when I do a numerical modelling of a case study in rock mechanics.