Question

in parallelogram vwxy, ( mangle wzx = 68^circ ), ( mangle vyw = 41^circ ), ( mangle wvx = 58^circ ), ( vw = 11 ), ( wz = 9 ), and ( xz = 3 ). find the length of ( overline{yz} ). note: the diagram is not drawn to scale.

Explanation:

Step1: Recall parallelogram diagonal property

In a parallelogram, the diagonals bisect each other. Wait, no, actually, in a parallelogram, the diagonals bisect each other? Wait, no, wait. Wait, in a parallelogram, the diagonals bisect each other? Wait, no, let's correct. Wait, in a parallelogram, the diagonals bisect each other? Wait, no, actually, in a parallelogram, the diagonals bisect each other. Wait, but here, we have triangle or something? Wait, no, looking at the diagram, in parallelogram VWXY, the diagonals are VX and WY, intersecting at Z? Wait, no, the diagram shows WZ = 9, XZ = 3, and we need YZ. Wait, in a parallelogram, the opposite sides are equal, and the diagonals bisect each other? Wait, no, wait, maybe I made a mistake. Wait, no, in a parallelogram, the diagonals bisect each other. Wait, but here, WZ is 9, and we need YZ. Wait, maybe the diagonals are WY and VX, intersecting at Z. Wait, in a parallelogram, the diagonals bisect each other, so WZ should equal YZ? Wait, no, that can't be, because WZ is 9, but maybe not. Wait, no, let's check the properties again. Wait, in a parallelogram, the diagonals bisect each other, so the midpoint of WY is Z, so WZ = YZ. Wait, that would mean YZ = WZ = 9? But that seems too easy. Wait, but maybe the diagram is different. Wait, no, the problem says "Find the length of YZ". Given that in a parallelogram, the diagonals bisect each other, so the point Z is the midpoint of WY. Therefore, WZ = YZ. Since WZ is 9, then YZ is also 9. Wait, but let's confirm. In a parallelogram, the diagonals bisect each other, so the intersection point of the diagonals is the midpoint of both diagonals. So if WY is a diagonal, then Z is the midpoint, so WZ = YZ. Therefore, YZ = WZ = 9.

Step1: Apply parallelogram diagonal property

In a parallelogram, diagonals bisect each other. So point \( Z \) is the midpoint of diagonal \( WY \). Thus, \( WZ = YZ \).
Given \( WZ = 9 \), we have \( YZ = WZ = 9 \).

Answer:

\( 9 \)