Geometry

Those are some nice AoPS Problems! Too bad they don't allow posting homework questions outside of the message board...

Problem #1  

Get the measure of the angles on either side of PAQ   

then subtract them from 90^o to leave the size of PAQ.  

tangent of BAP is 5 / 7  =  0.7143   

tangent of DAQ is 5 / 7  =  0.7143  

note that tan^–1(0.7143)  =  35.54^o  

BAP + DAQ  =  35.54^o + 35.54^o  =  71.08^o  

PAQ  =  90^o – 71.08^o  =  18.92^o  

sin(18.92^o)  =  0.324  

_.