1. **State the problem:** We have a right triangle $\triangle PQR$ with a right angle at $R$. Point $S$ lies on segment $PQ$ such that two smaller right triangles are formed inside the larger triangle. We need to complete the table relating segments and their geometric means. 2. **Recall the geometric mean theorem for right triangles:** If $S$ is the foot of the altitude from the right angle $R$ to the hypotenuse $PQ$, then: - The altitude $RS$ is the geometric mean of the two segments it divides the hypotenuse into: $PS$ and $SQ$. - Each leg of the triangle is the geometric mean of the hypotenuse and the segment of the hypotenuse adjacent to that leg. 3. **Apply the theorem to the segments:** - $RS = \sqrt{PS \times SQ}$ (altitude is geometric mean of $PS$ and $SQ$) - $PR = \sqrt{PQ \times PS}$ (leg $PR$ is geometric mean of $PQ$ and $PS$) - $QR = \sqrt{PQ \times SQ}$ (leg $QR$ is geometric mean of $PQ$ and $SQ$) 4. **Complete the table:** - For segments $PS$ and $QS$, the geometric mean is $RS$. - For segments $PQ$ and $QS$, the geometric mean is $QR$. - For segments $PS$ and $PQ$, the geometric mean is $PR$ (given in the table). 5. **Final completed table:** | Segments | Geometric Mean | |----------------|----------------| | PS and QS | $RS$ | | PS and PQ | $PR$ | | PQ and QS | $QR$ | This uses the properties of the right triangle altitude and the geometric mean relationships.