1. **Problem statement:** Given kite ABCD with BN = 15 and AB = 17, find the length of diagonal BD. 2. **Understanding the kite properties:** In a kite, two pairs of adjacent sides are equal. Also, the diagonals intersect at right angles. 3. **Using the right triangle formed:** Since BN is a segment from B to the intersection point N of the diagonals, and AB is a side of the kite, triangle ABN is right-angled at N. 4. **Apply the Pythagorean theorem:** In right triangle ABN, $$AB^2 = AN^2 + BN^2$$ Given $AB=17$ and $BN=15$, substitute: $$17^2 = AN^2 + 15^2$$ $$289 = AN^2 + 225$$ 5. **Solve for AN:** $$AN^2 = 289 - 225 = 64$$ $$AN = \sqrt{64} = 8$$ 6. **Find BD:** Since BD is the full diagonal and N is the midpoint of BD, $$BD = 2 \times BN = 2 \times 15 = 30$$ **Final answer:** $$\boxed{30}$$