1. **Problem statement:** Given kite ABCD with BN = 15 and AB = 17, find the length of diagonal BD. 2. **Understanding the kite properties:** In kite ABCD, diagonals intersect at right angles. BN is perpendicular to AC at point N. 3. **Given data:** - BN = 15 - AB = 17 - AN = 4.6 - NC = 6 4. **Goal:** Find BD, which is the length from B to D. 5. **Step 1: Use right triangle ABN to find BN and AN relationship:** Since AB = 17 and BN = 15, use the Pythagorean theorem in triangle ABN: $$AB^2 = AN^2 + BN^2$$ 6. **Step 2: Calculate AN using Pythagorean theorem:** $$17^2 = AN^2 + 15^2$$ $$289 = AN^2 + 225$$ $$AN^2 = 289 - 225 = 64$$ $$AN = \sqrt{64} = 8$$ 7. **Step 3: Check given AN:** The problem states AN = 4.6, but calculation shows AN = 8. This suggests a discrepancy or that AN = 4.6 is approximate or for another segment. We will proceed with the calculated AN = 8 for accuracy. 8. **Step 4: Find diagonal AC:** $$AC = AN + NC = 8 + 6 = 14$$ 9. **Step 5: Use right triangle BND to find BD:** Since BN is perpendicular to AC at N, and BD is the diagonal from B to D, BD can be found by: $$BD = 2 \times BN = 2 \times 15 = 30$$ This is because in a kite, one diagonal is bisected by the other at right angles. **Final answer:** $$\boxed{BD = 30}$$