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. 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, we can find LY: $$LY = \sqrt{AY^2 + AL^2} = \sqrt{7^2 + 3.5^2} = \sqrt{49 + 12.25} = \sqrt{61.25}$$ 4. **Calculate LY:** $$LY = 7.83 \text{ (rounded to two decimal places)}$$ 5. **Finding LW:** Since LW = AL + AW, and AW = LY (because of the right angles and the way the triangle is constructed), then: $$LW = AL + LY = 3.5 + 7.83 = 11.33$$ 6. **Final answer:** $$LW = 11.33 \text{ units}$$ This is the length of segment LW rounded to the nearest hundredth.