1. **State the problem:** Given a right triangle with points J, L, K, and point M on segment JK, with JM = 4 and JK = 9, find the length LK. 2. **Understand the setup:** Since M lies on JK, and JM = 4, JK = 9, then MK = JK - JM = 9 - 4 = 5. 3. **Use the right angle properties:** The problem states right angles at L and M, indicating that LM is perpendicular to JK. 4. **Apply the geometric mean theorem (right triangle altitude theorem):** In a right triangle, the altitude to the hypotenuse creates two segments on the hypotenuse, and the length of each leg is the geometric mean of the hypotenuse and the adjacent segment. Specifically, if LM is the altitude from L to hypotenuse JK, then: $$ JM \times MK = LM^2 $$ and $$ JL^2 = JM \times JK $$ Similarly, $$ LK^2 = MK \times JK $$ 5. **Calculate LK:** Given JK = 9, MK = 5, $$ LK^2 = MK \times JK = 5 \times 9 = 45 $$ 6. **Find LK:** $$ LK = \sqrt{45} = \sqrt{9 \times 5} = 3\sqrt{5} $$ 7. **Final answer:** $$ \boxed{LK = 3\sqrt{5}} $$