If the quadrilaterals are similar, the ratio of corresponding side lengths will be the same. Therefore, let's calculate the length of the polygons sides with the Distance Formula. In doing so, we can also investigate what type of quadrilateral these are.
Side
Points
sqrt((x_2-x_1)^2+(y_2-y_1)^2)
d
RQ
( - 2,2)
( - 1,0)
sqrt(( - 2-( - 1))^2+( 2- 0)^2)
sqrt(5)
QT
( 2,1)
( - 1,0)
sqrt(( 2-( - 1))^2+( 1- 0)^2)
sqrt(10)
TS
( 2,1)
( 1,3)
sqrt(( 2- 1)^2+( 1- 3)^2)
sqrt(5)
SR
( 1,3)
( - 2,2)
sqrt(( 1-( - 2))^2+( 3- 2)^2)
sqrt(10)
XW
( 0,2)
( 4,4)
sqrt(( 0- 4)^2+( 2- 4)^2)
sqrt(20)
WZ
( 0,2)
( 2,- 4)
sqrt(( 0- 2)^2+( 2-( - 4))^2)
sqrt(40)
ZY
( 2,-4)
( 6,- 2)
sqrt(( 2- 6)^2+( - 4-( - 2))^2)
sqrt(20)
YX
( 6,-2)
( 4,4)
sqrt(( 6- 4)^2+( - 2- 4)^2)
sqrt(40)
Since opposite sides in each quadrilateral have equal lengths, we know that they must be parallelograms. If these parallelograms are similar, the ratio of corresponding sides has to be the same.
sqrt(40)/sqrt(10)? =sqrt(20)/sqrt(5) ⇔ 2 = 2
Since the ratio between corresponding sides is the same, we know that the parallelograms are similar.
Transformations
The parallelograms have different orientations, positions, and size. Therefore, to make the polygons map onto each other, we have to rotate one of them, translate it so that two corresponding vertices map onto each other, and then dilate. First, we will rotate the parallelograms to have the same orientation.
Next, we will translate the rotated polygon so that Q' and W map onto each other.
Finally, we will have to dilate R'S''T''Q'' by a factor of 2 to make it the same size as WXYZ
