Question

In parallelogram ABCD, AB = 16 cm, DA = 3="\sqrt{2}" align="absmiddle" class="latex-formula"> cm, and sides AB and DA form a 45-degree interior angle. In isosceles trapezoid WXYZ with WX ≠ YZ, segment WX is the longer parallel side and has length 16 cm, and two interior angles each have a measure of 45 degrees. Trapezoid WXYZ has the same area as parallelogram ABCD. What is the length of segment YZ?

Answer 1

Answer:

length of segment YZ is 8 cm

Step-by-step explanation:

given data

AB = 16 cm

DA = 3$\sqrt{2}$ cm

AB and DA form interior angle  =  45-degre

WX ≠ YZ

WX = 16 cm

to find out

length of segment YZ

solution

area of △ABD is  the same as the area of △BCD

and

area of △ABD is express as

area of △ABD = AB × AD × sin(45) ÷ 2    ............1

put here value

area of △ABD = 16 × 3√2 × sin(45) ÷ 2

area of △ABD = 24

and

area of the parallelogram is

area of the parallelogram = 24 × 2

area of the parallelogram = 48

so

now we will consider here YZ = x

and Since ZY XW is isosceles trapezoid

so here we can say that

WM = ZM = (16 - x)  ÷ 2   .......................2

so area of  trapezoid will be

area of  trapezoid = $\frac{ZY + WX }{2} \times ZM$         .......................3

area of  trapezoid = $\frac{x+16}{2} \times \frac{16 - x}{2}$    

48 = $\frac{x+16}{2} \times \frac{16 - x}{2}$  

solve it we get

x = 8

so length of segment YZ is 8 cm