Question

19 rectangles zyxw and zyxw are shown in the coordinate plane below. which statement describes a sequence of transformations that can be used to verify that rectangle zyxw is similar to rectangle zyxw? a. dilate rectangle zyxw with a scale factor of \\( \frac { 1 } { 2 } \\) about the origin, followed by a translation 14 units up and 12 units to the left. b. dilate rectangle zyxw with a scale factor of \\( \frac { 1 } { 2 } \\) about the origin, followed by a reflection across the y - axis. c. dilate rectangle zyxw with a scale factor of 2 about the origin, followed by a \\( 180 ^ { \circ } \\) rotation about the origin. d. dilate rectangle zyxw with a scale factor of 2 about the origin, followed by a translation 14 units up and 12 units to the left.

Explanation:

Step1: Analyze the size relationship

Rectangle \(Z'Y'X'W'\) is larger than rectangle \(ZYXW\). If we assume the side - length of rectangle \(ZYXW\) is \(a\) and \(b\), and the side - length of rectangle \(Z'Y'X'W'\) is \(A\) and \(B\). By counting the grid units, we find that the scale factor \(k=\frac{A}{a}=\frac{B}{b} = 2\) (for example, if the length of \(ZW\) in rectangle \(ZYXW\) is \(2\) units and the length of \(Z'W'\) in rectangle \(Z'Y'X'W'\) is \(4\) units). So we should dilate rectangle \(ZYXW\) with a scale factor of \(2\) about the origin.

Step2: Analyze the translation

After dilation, we need to move the dilated rectangle. By counting the vertical and horizontal distances between the corresponding points (e.g., between the center of the two rectangles or corresponding vertices). The vertical distance ( \(y\) - direction) is \(14\) units up and the horizontal distance (\(x\) - direction) is \(12\) units to the left.

Answer:

D. Dilate rectangle \(ZYXW\) with a scale factor of \(2\) about the origin, followed by a translation \(14\) units up and \(12\) units to the left.