How can i select tools and techniques to analyze the relationship among the sides of a right triangle

If it is a special triangle (either 45-45-90 or 30-60-90) then you select the tools from the rules of special triangles.  

Otherwise, if you know two angles and a side of the triangle (AAS or ASA) you use law of sines.  If you know two sides and an angle (SSA) you can also use law of sines.

Use the law of cosines when you know two sides and the angle in between them (SAS).