Question

which composition of similarity transformations maps polygon abcd to polygon abcd? a dilation with a scale factor greater than 1 and then a reflection a dilation with a scale factor less than 1 and then a reflection a dilation with a scale factor greater than 1 and then a translation a dilation with a scale factor less than 1 and then a translation

Explanation:

Step1: Analyze Dilation Scale Factor

Compare the size of polygon \(ABCD\) and \(A'B'C'D'\). \(A'B'C'D'\) is smaller than \(ABCD\), so the dilation scale factor is less than 1.

Step2: Analyze Transformation Type After Dilation

Check the orientation: \(ABCD\) and \(A'B'C'D'\) are mirror - images (reflection) or just translated? The shapes are similar in orientation after considering the size change, but the key is the size (scale factor < 1) and then a reflection? Wait, no, actually, looking at the positions, after dilation (scale factor < 1) to make it smaller, then a translation? Wait, no, the correct approach: First, dilation: since \(A'B'C'D'\) is smaller, scale factor < 1. Then, check the transformation. The original polygon \(ABCD\) and \(A'B'C'D'\) – the direction of the sides: after dilation (making it smaller), we need to see if it's a reflection or translation. Wait, the correct option is a dilation with scale factor less than 1 and then a translation? Wait, no, let's re - examine. Wait, the first step: dilation scale factor. \(ABCD\) is larger, \(A'B'C'D'\) is smaller, so scale factor < 1. Then, the transformation from the dilated (smaller) version of \(ABCD\) to \(A'B'C'D'\) – is it a reflection or translation? Wait, the correct option is "a dilation with a scale factor less than 1 and then a translation"? Wait, no, the options: let's list the options again.

Option 1: dilation (scale > 1) + reflection

Option 2: dilation (scale < 1) + reflection

Option 3: dilation (scale > 1) + translation

Option 4: dilation (scale < 1) + translation

Wait, the correct analysis: The polygon \(A'B'C'D'\) is smaller than \(ABCD\), so dilation scale factor is less than 1. Then, the transformation from the dilated (smaller) \(ABCD\) to \(A'B'C'D'\) – looking at the coordinates, the orientation: \(ABCD\) and \(A'B'C'D'\) – the key is that after making it smaller (dilation scale < 1), we need to translate? Wait, no, the correct answer is the fourth option? Wait, no, let's check the graph again.

Wait, the original polygon \(ABCD\): points \(A\), \(B\), \(C\), \(D\). The image \(A'B'C'D'\) is smaller. So dilation with scale factor less than 1 (to reduce size) and then a translation? Wait, no, the correct option is "a dilation with a scale factor less than 1 and then a translation"? Wait, the options:

Wait, the fourth option is "a dilation with a scale factor less than 1 and then a translation". Let's verify:

1. Dilation scale factor: Since \(A'B'C'D'\) is smaller, scale factor < 1.

1. After dilation (making \(ABCD\) smaller to the size of \(A'B'C'D'\) before transformation), then we translate it to the position of \(A'B'C'D'\). Wait, but also, is there a reflection? No, the orientation (the direction of the polygon) – \(ABCD\) and \(A'B'C'D'\) – if we dilate \(ABCD\) with scale factor < 1 (making it smaller) and then translate it, we get \(A'B'C'D'\). Wait, but the options:

Wait, the correct answer is the fourth option: "a dilation with a scale factor less than 1 and then a translation". Wait, no, I think I made a mistake earlier. Let's re - evaluate.

The first step: Dilation scale factor. \(A'B'C'D'\) has a smaller area (or side length) than \(ABCD\), so scale factor is less than 1.

The second step: After dilation (scale < 1), the transformation from the dilated \(ABCD\) to \(A'B'C'D'\) is a translation (moving the dilated figure to the position of \(A'B'C'D'\)) rather than a reflection. So the correct option is "a dilation with a scale factor less than 1 and then a translation".

Answer:

a dilation with a scale factor less than 1 and then a translation (the fourth option in the list of options provided)