Explanation:

Triangle PQR is an equilateral triangle and PS is abisector of angle RPQ
therefore, angle PRQ=angle RPQ=60 and angleRPS=angleQPS=30
Again as  angle RPS and angle RQS are lie in the same segmant of the same circle
therefore angle RPS=angle RQS =30
angleTRQ=180-angle PRQ=180-60=120
in triangle RTQ, we have 
angle RTQ +angle TQR + angle TRQ=180
angle RTQ + 120+ 30=180
therefore, angle RTQ=180-150=30
therefore , we can say triangle RTQ is an isoceles triangle whose RQ=RT
therefore,  RT:RQ=1:1