1. **State the problem:** We have a right triangle $\triangle PQR$ with right angle at $R$. A segment $QS$ is drawn perpendicular to $PR$, intersecting at $S$. We need to complete the table of geometric means for segments. 2. **Recall the geometric mean theorem:** In a right triangle, the altitude to the hypotenuse creates two smaller right triangles similar to the original. The altitude is the geometric mean between the two segments it divides the hypotenuse into. 3. **Identify segments:** - $PR$ is the hypotenuse. - $QS$ is the altitude to the hypotenuse. - $PS$ and $SR$ are the two segments into which $S$ divides $PR$. 4. **Apply the geometric mean relationships:** - $QS$ is the geometric mean of $PS$ and $SR$: $$QS = \sqrt{PS \times SR}$$ - $PR$ is the geometric mean of $PQ$ and $PR$: This is incorrect; the correct relation is: - $PQ$ is the geometric mean of $PR$ and $PS$: $$PQ = \sqrt{PR \times PS}$$ - $QR$ is the geometric mean of $PR$ and $SR$: $$QR = \sqrt{PR \times SR}$$ 5. **Complete the table:** - For segments $PS$ and $QS$, the geometric mean is $QS$. - For segments $PQ$ and $QS$, the geometric mean is $PS$. - For segments $SR$ and $QS$, the geometric mean is $SR$. 6. **Final table:** | Segments | Geometric Mean | |-------------------|----------------| | $PS$ and $SR$ | $QS$ | | $PQ$ and $PR$ | $PS$ | | $QR$ and $PR$ | $SR$ | **Note:** The problem's table had blanks; the correct pairs and means are as above based on the altitude theorem.