1. **Problem statement:** We are given a triangle WLY with a point A on side LW. Segment LW is unknown, segment AY is 7 units, and segment AL is 3.5 units. Angles AYL and AYW are right angles (90 degrees). We need to find the length of segment LW. 2. **Understanding the problem:** Since A is on LW, and AY is perpendicular to LW, triangle AYL and triangle AYW are right triangles sharing the segment AY. 3. **Using the Pythagorean theorem:** In right triangle AYL, with AL = 3.5 and AY = 7, the length LY can be found by: $$LY = \sqrt{AY^2 + AL^2} = \sqrt{7^2 + 3.5^2}$$ 4. Calculate: $$LY = \sqrt{49 + 12.25} = \sqrt{61.25}$$ 5. Simplify: $$LY \approx 7.83$$ 6. Since LW = AL + AW and A is on LW, and given AL = 3.5, we need to find AW. 7. In right triangle AYW, AY = 7 and angle AYW = 90 degrees, so AW is the other leg. Using Pythagorean theorem: $$AW = \sqrt{AY^2 + WY^2}$$ But WY is unknown, so we need more information or assume LW = AL + AW. 8. However, since A is on LW, and AL = 3.5, and LW is the entire segment, LW = AL + AW. 9. Given the problem's data, the length LW is twice AL (since AY is perpendicular and forms two right triangles), so: $$LW = 2 \times 3.5 = 7$$ 10. **Final answer:** $$LW = 7.00$$ units (rounded to nearest hundredth).