A dilation is a transformation in which a figure is enlarged or reduced with respect to a fixed point.
Next we have to determine which of these transformations were used to map quadrilateral ABCD to quadrilateral QRST. Let's consider the given diagram.
We can see that AD is vertical and is located in the second quadrant, and TQ is horizontal and is located in the first quadrant. This suggests a rotation counterclockwise about the origin of 270^(∘). Recall that the coordinate rule for such rotation can be written as (x,y)→ (y,- x).
Quadrilateral QRST appears to be about half as large as quadrilateral A'B'C'D'. Therefore, our second transformation is a dilation centered at the origin by a scale factor of 12. Recall that the coordinate rule for such a dilation can be written as (x,y)→ ( 12x, 12y).
Finally, we will write the coordinate rule for the composition of transformations that maps quadrilateral ABCD to quadrilateral QRST. Recall that the first transformation is (x,y)→(y, - x) and the second transformation is (x,y)→( 12x, 12y). Let's write the composition of these two transformations.
(x,y)→(y, -x)→( 1/2y, - 1/2x)
⇕
(x,y)→(1/2y,-1/2x)
b Recall that two figures are similar if and only if there is a similarity transformation that maps one of the figures onto the other. In other words, similar figures have the same shape but not necessarily the same size. Let's consider the given diagram.
From Part A we know that a composition of a rotation and a dilation maps quadrilateral ABCD to quadrilateral QRST. Rotations and dilations are similarity transformations. A composition of similarity transformations is also a similarity transformation. In conclusion, quadrilaterals ABCD and QRST are similar.
