1. **Problem:** Find the length of side JK in the right triangle JKL where angle L is 45°, side KL = 42, and JL is unknown. 2. **Given:** - Right triangle with right angle at J - Angle L = 45° - Side KL = 42 - Using the Pythagorean theorem: $$42^2 = JK^2 + JL^2$$ 3. **Step 1: Write the Pythagorean theorem** $$KL^2 = JK^2 + JL^2$$ 4. **Step 2: Substitute known values** $$42^2 = JK^2 + JL^2$$ $$1764 = JK^2 + JL^2$$ 5. **Step 3: Use the property of 45°-45°-90° triangle** In a 45°-45°-90° triangle, the legs are equal, so: $$JK = JL$$ 6. **Step 4: Substitute $JL = JK$ into the equation** $$1764 = JK^2 + JK^2 = 2JK^2$$ 7. **Step 5: Solve for $JK^2$** $$JK^2 = \frac{1764}{2} = 882$$ 8. **Step 6: Find $JK$ by taking the square root** $$JK = \sqrt{882} = \sqrt{9 \times 98} = 3\sqrt{98} = 3\sqrt{49 \times 2} = 3 \times 7 \sqrt{2} = 21\sqrt{2}$$ **Final answer:** $$JK = 21\sqrt{2}$$