In this diagram, we are dealing with the construction of a tangent to a circle from a point on the circle. The problem is asking us to identify the correct statement about the points and segments depicted.

Analysis of Each Option:

1. Option A: $\overline{PQ} \cong \overline{PR}$
   This option suggests that segment $PQ$ is congruent to segment $PR$. In this configuration, $PQ$ and $PR$ are not necessarily equal because $PQ$ is a radius of the circle (connecting center $O$ to a point $Q$ on the circle), while $PR$ is an external tangent segment and typically would not be congruent to a radius.

2. Option B: $OQ + QP = QP + PR$
   This statement does not hold meaningfully in this configuration because it is merely rearranging the terms. The equality it suggests doesn’t reflect any particular geometric property.

3. Option C: $P$ is the midpoint of $\overline{OR}$
   This option suggests that point $P$ is the midpoint of $OR$. However, in the diagram, $P$ is a point on the tangent line, not necessarily the midpoint of $OR$.

4. Option D: $\overline{OP} \perp \overline{QR}$
   This is likely to be true, as $\overline{OP}$ is perpendicular to the tangent line $\overline{QR}$ at the point of tangency $P$. By definition, a radius drawn to the point of tangency is perpendicular to the tangent line.

Conclusion

The correct answer is:

• Option D: $\overline{OP} \perp \overline{QR}$, as the radius $OP$ is perpendicular to the tangent line $QR$ at the point of tangency $P$.

Would you like further details or explanations on tangent properties or related geometry concepts?

Here are five follow-up questions that may deepen your understanding:

1. Why is a radius always perpendicular to a tangent at the point of tangency?
2. How can you construct a tangent to a circle from a point outside the circle?
3. What are the properties of tangent segments drawn from a common external point?
4. How can the midpoint of a line segment be determined geometrically?
5. How do congruent segments differ from perpendicular segments in geometric constructions?

Tip: In geometry, remember that a tangent to a circle at any point is perpendicular to the radius drawn to that point.