To find the measures of segments $OP$, $QR$, and $QT$, let’s analyze the information provided in the image.

Given Information:

1. QT is a diameter of the circle that passes through point $S$.
2. $\triangle OQT$ is a right triangle with $\angle OQT = 90^\circ$.
3. The length of $OT = 12$.
4. The length of $PR = 5$, and segments $OR$ and $QR$ are both perpendicular to $QT$.

Let’s solve for each segment step-by-step.

1. Measure of $OP$:

Since $\triangle OQR$ is a right triangle with $\angle OQR = 90^\circ$, $OP$ can be calculated by adding $OR$ and $PR$: $OP = OR + PR = OR + 5.$ However, $OR$ is unknown, and we would need further information about the exact length of $OR$ to determine $OP$ accurately.

2. Measure of $QR$:

Similarly, $QR$ is part of the right triangle $\triangle OQR$. To find $QR$, we would need additional information about either the radius of the circle or other specific segment lengths within this triangle.

3. Measure of $QT$:

Since $QT$ is the diameter of the circle and extends from one end of the circle to the other, we can deduce that: $QT = 2 \times OT = 2 \times 12 = 24.$

Summary of Findings:

• $OP$: We need the length of $OR$ to determine $OP$ accurately.
• $QR$: Requires additional data to solve.
• $QT = 24$, since it is twice the radius (given that $OT = 12$).

Would you like further details, or have any specific questions about this setup?

Further Questions to Explore:

1. What additional information would allow us to solve for $OP$ and $QR$ exactly?
2. Can the Pythagorean theorem be applied to find any missing sides within this configuration?
3. How would knowing the radius of the circle help in solving for unknown segments?
4. Are there other ways to confirm that $\angle OQT$ is indeed $90^\circ$ based on circle properties?
5. Could trigonometric relationships be used in this setup to solve for missing segments?

Tip:

In a circle, any triangle formed with the diameter as one of its sides will always be a right triangle. This is a result of the Thales' theorem.