1. **Problem statement:** We have a right triangle EHL with a right angle at E. A segment EP is perpendicular to the hypotenuse HL, with EP = 9.7 units, PH = 18.6 units, and EH = 21 units. We need to find the length EL. 2. **Key information:** In a right triangle, the altitude to the hypotenuse creates two smaller right triangles similar to the original triangle and to each other. 3. **Formula used:** The altitude EP satisfies the relation $$EP^2 = PH \times PL$$ where PL is the segment of the hypotenuse from point P to L. 4. We know the entire hypotenuse HL = PH + PL. We have PH = 18.6 and HL = EH = 21 (since EH is the hypotenuse). 5. Calculate PL: $$PL = HL - PH = 21 - 18.6 = 2.4$$ 6. Use the altitude formula: $$EP^2 = PH \times PL$$ $$9.7^2 = 18.6 \times 2.4$$ Calculate left side: $$9.7^2 = 94.09$$ Calculate right side: $$18.6 \times 2.4 = 44.64$$ Since these are not equal, check if EH is the hypotenuse or if HL is the hypotenuse. 7. Given EH = 21, and PH + PL = HL, HL must be the hypotenuse. So HL = PH + PL = 18.6 + PL. 8. Use the similarity property: $$EH^2 = EL \times HL$$ We know EH = 21, HL = 18.6 + PL, and EL is unknown. 9. Also, from the right triangle properties: $$EL = \sqrt{EH^2 - HL^2}$$ but HL is unknown. 10. Use the relation from the altitude: $$EP^2 = PH \times PL$$ $$94.09 = 18.6 \times PL$$ Solve for PL: $$PL = \frac{94.09}{18.6} \approx 5.06$$ 11. Now HL = PH + PL = 18.6 + 5.06 = 23.66 12. Use Pythagoras theorem in triangle EHL: $$EL = \sqrt{EH^2 - HL^2} = \sqrt{21^2 - 23.66^2}$$ Calculate squares: $$21^2 = 441$$ $$23.66^2 \approx 559.56$$ Since $441 - 559.56$ is negative, this is impossible, so re-examine the triangle sides. 13. Since EH = 21 and HL = 18.6 + PL, and EP is altitude to HL, HL is the hypotenuse, so EH and EL are legs. 14. Use the similarity property: $$EH^2 = PH \times HL$$ $$21^2 = 18.6 \times HL$$ Calculate: $$441 = 18.6 \times HL$$ Solve for HL: $$HL = \frac{441}{18.6} = 23.71$$ 15. Now find EL using: $$EL^2 = PL \times HL$$ We have PL = HL - PH = 23.71 - 18.6 = 5.11 Calculate: $$EL^2 = 5.11 \times 23.71 = 121.1$$ 16. Find EL: $$EL = \sqrt{121.1} \approx 11.0$$ **Final answer:** $$EL \approx 11.0$$ units (rounded to the nearest tenth).