The intersecting tangents theorem says that the measure of angle PQR is half the difference of the measures of the arcs that the tangents intercept. This means
[tex]m\angle PQR=\dfrac{m\widehat{PR}_{\rm maj}-m\widehat{PR}_{\rm min}}2[/tex]
where [tex]\widehat{PR}_{\rm maj}[/tex] refers to the major arc PR, and [tex]\widehat{PR}_{\rm min}[/tex] refers to the minor arc PR (the one you know has measure [tex](100x)^\circ[/tex]).
The major and minor arcs PR form the circle, so their measures add up to 360º:
[tex]m\widehat{PR}_{\rm maj}=360^\circ-(100x)^\circ[/tex]
Then by the aforementioned theorem, we have
[tex](81x-1)^\circ=\dfrac{\left(360^\circ-(100x)^\circ\right)-(100x)^\circ}2[/tex]
[tex]81x-1=\dfrac{360-200x}2[/tex]
[tex]81x-1=180-100x[/tex]
[tex]181x=181[/tex]
[tex]\implies x=1[/tex]
Then the minor arc PR has measure 100º, making the answer A.
