# Which composition of similarity transformations maps polygon abcd to polygon a’b’c’d’?

A. a dilation with a scale factor less than 1 and then a reflection

B. a dilation with a scale factor less than 1 and then a translation

C. a dilation with a scale factor greater than 1 and then a reflection

D. a dilation with a scale factor greater than 1 and then a translation

Option A is correct.

To map polygon ABCD to polygon A’B’C’D’ using a composition of similarity transformations, we need to apply two transformations in sequence: a dilation with a scale factor less than 1 and a reflection.

A dilation with a scale factor less than 1 reduces the size of the polygon by the same factor in all directions. This means that the distance between any two points on the polygon is decreased by the same factor.

Next, a reflection flips the polygon over a line of reflection, which is a line that acts as a mirror, reflecting the polygon across it.

By applying these two transformations in sequence, we first reduce the size of the polygon and then reflect it over a line of reflection. This maps the original polygon ABCD to a new polygon A’B’C’D’, which is similar to the original polygon but smaller and reflected.

## Apply:

Let’s say that the original polygon ABCD has vertices at points A(2,2), B(2,4), C(4,4), and D(4,2) in the Cartesian plane.

To transform this polygon to A’B’C’D’, we will use a dilation with a scale factor less than 1 and a reflection over the line y = x.

First, we apply the dilation with a scale factor less than 1. Let’s say we use a scale factor of 1/2. This means that every point on the original polygon ABCD will be moved half the distance towards the origin. The resulting polygon will have vertices at points A'(1,1), B'(1,2), C'(2,2), and D'(2,1).

Next, we apply a reflection over the line y = x. This means that every point on the polygon will be reflected across this line, so the x and y coordinates of each point will be swapped. The resulting polygon A’B’C’D’ will have vertices at points A'(1,1), B'(2,1), C'(2,2), and D'(1,2).

Therefore, the composition of similarity transformations that maps polygon ABCD to polygon A’B’C’D’ is a dilation with a scale factor less than 1 and a reflection over the line y = x.

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