Two-dimensional (2D) crystals exhibit unique and exceptional properties and show promise for various applications. In this work, we systematically studied the structures of a 2D boronphosphide (BP) monolayer with different stoichiometric ratios (BPx, x = 1, 2, 3, 4, 5, 6 and 7) and observed that each compound had a stable 2D structure with metallic or semiconducting electronic properties. Surprisingly, for the BP5 compounds, we discovered a rare penta-graphene-like 2D structure with a tetragonal lattice. This monolayer was a semiconductor with a quasi-direct band gap of 2.68 eV. More importantly, investigation of the strain effect revealed that small uniaxial strain can trigger the band gap of the penta-BP5 monolayer to transition from a quasi-direct to direct band gap, whereas moderate biaxial strain can cause the penta-BP5 to transform from a semiconductor into a metal, indicating the great potential of this material for nanoelectronic device applications based on strain-engineering techniques. The wide and tuneable band gap of monolayer penta-BP5 makes it more advantageous for high-frequency-response optoelectronic materials than the currently popular 2D systems, such as transition metal dichalcogenides and black phosphorus. These unique structural and electronic properties of 2D BP sheets make them promising for many potential applications in future nanodevices.