Question

what is the perimeter of parallelogram wxyz? the perimeter is \\(\square\\) (type an integer or a decimal.)

Explanation:

Step1: Identify parallelogram properties

In a parallelogram, opposite sides are equal. Also, the diagonals bisect each other, but here we see sides: \( WZ = 42 \), and from the markings, \( WX \) can be found? Wait, no, wait. Wait, the figure: WXYZ is a parallelogram? Wait, no, actually, looking at the diagonals: in a parallelogram, diagonals bisect each other, but here the triangles formed by diagonals have sides 29, 29, and 42? Wait, no, maybe it's a parallelogram with sides: \( WZ = 42 \), and \( WX \): wait, the diagonals are equal? Wait, no, the markings: the sides with one red mark are equal, two red marks are equal. Wait, actually, WXYZ: let's check the sides. \( WZ = 42 \), and \( XY = WZ = 42 \) (opposite sides of parallelogram). Then, what about \( WX \) and \( YZ \)? Wait, the triangles: triangle \( WXZ \) has sides? Wait, no, the diagonals: in a parallelogram, diagonals bisect each other, but here the length from W to the intersection and X to intersection is 29? Wait, no, the segments of the diagonals are 29, so the diagonals are equal? Wait, if diagonals are equal, then it's a rectangle? Wait, maybe it's a rectangle (a type of parallelogram) with length 42 and width: wait, no, let's re-examine. Wait, the side \( WZ = 42 \), and the other side: let's see, the diagonals are equal (both diagonals are \( 29 + 29 = 58 \))? Wait, no, in a rectangle, diagonals are equal, and sides: if \( WZ = 42 \), and diagonal is 58, then we can find the other side using Pythagoras: \( a^2 + b^2 = c^2 \), where \( c = 58 \), \( a = 42 \), so \( b = \sqrt{58^2 - 42^2} \). Let's calculate that: \( 58^2 = 3364 \), \( 42^2 = 1764 \), so \( 3364 - 1764 = 1600 \), so \( b = \sqrt{1600} = 40 \). Wait, but wait, the figure: WX is the other side. Wait, maybe I made a mistake. Wait, the problem is to find the perimeter of parallelogram WXYZ. So perimeter of parallelogram is \( 2 \times (length + width) \). So we need two adjacent sides. \( WZ = 42 \), and \( WX \): let's find \( WX \). Wait, the diagonals: in the figure, the diagonals are split into 29 each, so diagonal length is \( 29 + 29 = 58 \). Then, in triangle \( WXZ \), we have sides \( WZ = 42 \), diagonal \( XZ = 58 \), and side \( WX \). Wait, no, \( XZ \) is a diagonal, so triangle \( WXZ \) has sides \( WX \), \( WZ = 42 \), and diagonal \( XZ = 58 \). Wait, no, that can't be, because \( 42^2 + WX^2 = 58^2 \)? Wait, no, \( 42^2 + 40^2 = 1764 + 1600 = 3364 = 58^2 \). Oh, right! So \( WX = 40 \). So the sides of the parallelogram are 42 and 40. Then perimeter is \( 2 \times (42 + 40) = 2 \times 82 = 164 \)? Wait, no, wait: wait, \( WZ = 42 \), \( WX = 40 \), so opposite sides: \( WZ = XY = 42 \), \( WX = YZ = 40 \). So perimeter is \( 2 \times (42 + 40) = 164 \). Wait, let's check again. Diagonal \( XZ = 58 \), side \( WZ = 42 \), so using Pythagoras: \( WX^2 + WZ^2 = XZ^2 \)? Wait, no, \( WX \) and \( WZ \) are adjacent sides, and \( XZ \) is the diagonal. So yes, \( WX^2 + 42^2 = 58^2 \). So \( WX^2 = 58^2 - 42^2 = (58 - 42)(58 + 42) = 16 \times 100 = 1600 \), so \( WX = 40 \). Then perimeter: \( 2 \times (42 + 40) = 2 \times 82 = 164 \).

Step2: Calculate perimeter

Perimeter of parallelogram = \( 2 \times (length + width) \) = \( 2 \times (42 + 40) \) = \( 2 \times 82 = 164 \).

Answer:

164