1. **State the problem:** We need to find the length of side JL in a right triangle JLK where angle L is 90°, angle K is 52°, and side KL is adjacent to angle K. We are given two cases for the length of KL: 3.4 mm and 7.6 mm. 2. **Formula and explanation:** In a right triangle, the tangent of an angle is the ratio of the opposite side to the adjacent side. Here, angle K is 52°, JL is opposite to angle K, and KL is adjacent to angle K. So, $$\tan(52^\circ) = \frac{JL}{KL}$$ Rearranging to find JL: $$JL = KL \times \tan(52^\circ)$$ 3. **Calculate $\tan(52^\circ)$:** Using a calculator, $$\tan(52^\circ) \approx 1.2799$$ 4. **Case a) KL = 3.4 mm:** $$JL = 3.4 \times 1.2799 = 4.3517$$ Rounded to 2 decimal places: $$JL \approx 4.35 \text{ mm}$$ 5. **Case b) KL = 7.6 mm:** $$JL = 7.6 \times 1.2799 = 9.7272$$ Rounded to 2 decimal places: $$JL \approx 9.73 \text{ mm}$$ **Final answers:** - a) JL = 4.35 mm - b) JL = 9.73 mm