1. **Problem statement:** We have a right triangle KPR with a right angle at K. A segment KL is perpendicular to the hypotenuse PR, dividing it into two segments PL and LR with lengths 7.5 and 11.7 units respectively. We need to find the length of KL. 2. **Formula used:** In a right triangle, the altitude to the hypotenuse (KL) satisfies the relation: $$KL = \sqrt{PL \times LR}$$ This comes from the geometric mean theorem, which states that the altitude to the hypotenuse is the geometric mean of the two segments it creates on the hypotenuse. 3. **Calculation:** $$KL = \sqrt{7.5 \times 11.7}$$ $$KL = \sqrt{87.75}$$ $$KL \approx 9.37$$ 4. **Rounding:** To the nearest tenth, $$KL \approx 9.4$$ 5. **Answer:** The length of segment KL is approximately 9.4 units.