1. **Problem Statement:** We have a kite-shaped quadrilateral CDEF with diagonals FD and EC intersecting at point B. We know that side ED is 85 and the segment DB (part of diagonal FD) is 77. We need to find the length of segment BE (part of diagonal EC). 2. **Properties of Kites:** In a kite, the diagonals are perpendicular, and one diagonal is bisected by the other. Specifically, the diagonal connecting the vertices where the two pairs of equal sides meet is bisected by the other diagonal. 3. **Applying the Property:** Since FD and EC intersect at B, and FD is bisected by EC, point B is the midpoint of FD. This means: $$FB = BD = 77$$ 4. **Using the Pythagorean Theorem:** Because the diagonals are perpendicular, triangle EBD is a right triangle with right angle at B. We know: - ED = 85 (hypotenuse) - BD = 77 (one leg) We want to find BE (the other leg). 5. **Calculate BE:** $$BE = \sqrt{ED^2 - BD^2} = \sqrt{85^2 - 77^2}$$ Calculate squares: $$85^2 = 7225$$ $$77^2 = 5929$$ Subtract: $$7225 - 5929 = 1296$$ Take the square root: $$BE = \sqrt{1296} = 36$$ 6. **Final Answer:** $$\boxed{36}$$ Thus, the length of segment BE is 36.