Studies of rare $B \rightarrow K\pi\mu\mu$ decays using the LHCb experiment

Within the Standard Model of particle physics, decays involving flavour-changing neutral current transitions cannot occur via tree-level Feynman diagrams and can only occur at loop level or higher. This leads to a heavy suppression of decay modes of this type and makes them sensitive to contributions from particles beyond the Standard Model that can enter in competing Feynman diagrams. By performing measurements of the rate and angular distribution of these decay modes, powerful tests of the Standard Model can be carried out. In this thesis, a first search for the decay $B^0_s \rightarrow \overline{K}{}^{*0}\mu^+\mu^-$ is presented using data samples collected by the LHCb collaboration during Run 1 and Run 2 of the LHC. An excess of this decay is found over the background-only hypothesis, providing first evidence for the decay mode with a significance of 3.4 standard deviations. A first measurement of the branching fraction of this decay is made and a discussion of the future prospects for measurements of the decay mode is presented.