The measure of the tangent line PQ to the circle is 6 units.
To solve for the length of PQ, we use the secant-tangent power theorem:
( tangent segment )² = External part of the secant segment × Secant segment.
Given that:
Tangent segment = PQ = ?
The external part of the secant segment = PR = 4
Secant segment = PR + RS = 4 + 5 = 9
Plug these values into the above formula and solve for the tangent line PQ:
( PQ )² = PR × ( PR + RS )
( PQ )² = 4 × ( 9 )
( PQ )² = 36
PQ = √36
PQ = 6 units
Therefore, segment PQ measures 6 units.