1. **Problem statement:** We have two braces placed against the side of a house forming angles of 45° and 30° with the vertical wall. The shorter brace is 8 m long at a 45° angle. We need to find the length of the longer brace which makes a 30° angle with the wall. 2. **Understanding the setup:** The wall is vertical, and the ground is horizontal, forming a right angle. The 8 m brace forms a 45° angle with the wall. The longer brace forms a 30° angle with the wall. 3. **Using trigonometry:** Since both braces start from the same point on the wall and end on the ground, they form right triangles with the wall and ground. 4. **Calculate the horizontal distance from the wall to the base of the 8 m brace:** Using the 45° brace: $$\text{horizontal distance} = 8 \times \sin 45^\circ = 8 \times \frac{\sqrt{2}}{2} = 8 \times 0.7071 = 5.6568\,m$$ 5. **Calculate the length of the longer brace using the 30° angle:** The longer brace forms a 30° angle with the wall, so its horizontal distance from the wall is: $$L \times \sin 30^\circ = L \times 0.5$$ Since both braces touch the ground at the same point, their horizontal distances must be equal: $$L \times 0.5 = 5.6568$$ 6. **Solve for $L$:** $$L = \frac{5.6568}{0.5} = \frac{5.6568}{\cancel{0.5}} \times \cancel{2} = 11.3136\,m$$ 7. **Final answer:** The length of the longer brace is approximately **11.3 m** (rounded to the nearest tenth).