1. **Problem Statement:** We have a kite-shaped quadrilateral VWXY with diagonals WY and VX intersecting at point Z. Given that segment YX = 5 and segment ZX = 4, we need to find the length of segment YZ. 2. **Properties of a Kite:** - The diagonals of a kite are perpendicular. - One diagonal (usually the one connecting the vertices between the two pairs of equal sides) is bisected by the other. - In kite VWXY, diagonals WY and VX intersect at Z, and Z is the midpoint of VX. 3. **Using the property that Z bisects VX:** Since Z is the midpoint of VX, segment VZ = ZX = 4. 4. **Using the Pythagorean theorem:** Since diagonals are perpendicular, triangle YZX is a right triangle with right angle at Z. We know YX = 5 and ZX = 4, so we can find YZ: $$YZ = \sqrt{YX^2 - ZX^2} = \sqrt{5^2 - 4^2} = \sqrt{25 - 16} = \sqrt{9} = 3$$ 5. **Answer:** The length of segment YZ is $3$.