The locus of all points from which a given segment subtends equal angles is a circle. Therefore:
[tex]RT\cdot RS=RQ\cdot RP[/tex]
so:
[tex]\begin{gathered} (4+6)(4)=(x+2+x)(x+2) \\ 10(4)=(2x+2)(x+2) \\ 40=2x^2+6x+4 \\ \end{gathered}[/tex]
Divide both sides by 2:
[tex]\begin{gathered} x^2+3x+2=20 \\ so\colon \\ x^2+3x-18=0 \end{gathered}[/tex]
The factors of -18 that sum to 3 are 6 and -3, therefore:
[tex]\begin{gathered} x^2+3x-18=(x+6)(x-3) \\ so\colon \\ x=3 \\ or \\ x=-6 \end{gathered}[/tex]
So:
[tex]\begin{gathered} x=3 \\ because\colon \\ RP>0 \\ RP=x+2 \\ RP=3+2 \\ RP=5 \end{gathered}[/tex]
