1. **Problem statement:** We are given a geometric figure with points B, K, C, L, and D. The segments BK = 10, KC = 5, BL = 13, and LD = y. We need to find the length y. 2. **Understanding the figure:** The figure consists of two connected triangles sharing some points. We can use the Pythagorean theorem or properties of triangles to find y. 3. **Assumption:** Since the problem involves segments BK, KC, BL, and LD, and the figure suggests right triangles, we assume triangles BKC and BLD are right triangles. 4. **Using the Pythagorean theorem on triangle BKC:** $$BC = BK + KC = 10 + 5 = 15$$ 5. **Using the Pythagorean theorem on triangle BLD:** Given BL = 13 and LD = y, if triangle BLD is right angled at L, then: $$BD^2 = BL^2 + LD^2 = 13^2 + y^2 = 169 + y^2$$ 6. **Relating BD to BC:** If points B, C, and D are collinear or related such that BD = BC = 15, then: $$BD = 15$$ 7. **Set up the equation:** $$BD^2 = 15^2 = 225$$ So, $$169 + y^2 = 225$$ 8. **Solve for y:** $$y^2 = 225 - 169 = 56$$ $$y = \sqrt{56} = \sqrt{4 \times 14} = 2\sqrt{14}$$ 9. **Final answer:** $$\boxed{y = 2\sqrt{14}}$$