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My rationale is to create a bigger square that embraces the given one. The points P, Q, R, and S touch the sides of the bigger square but not bisect them! They share each side of the bigger square by 4:3; this fact is clear from the information that PT is perpendicular to QT, QT=3, and PT=4.
Further, you have to see that the right vertical side of the bigger square is 7, and the bottom one is shared 4:3, maiking it possible to find RT via the right triangle (7-3-RT).
The trick is that a small square is inscribed into a bigger one, not bysecting the sides of the latter, but splitting them into a 4:3 proportion.