Question

two trapezoids are represented on the coordinate plane. which series of transformations of one of the trapezoids can be used to justify that trapezoid a and trapezoid b are similar? a dilation by scale factor 2 centered at the origin followed by a reflection across the y - axis a dilation by scale factor 2 centered at the origin followed by a 90 - degree rotation counterclockwise centered at the origin a dilation by scale factor \\( \frac { 1 } { 2 } \\) centered at the origin followed by a reflection across the y - axis a dilation by scale factor \\( \frac { 1 } { 2 } \\) centered at the origin followed by a 90 - degree rotation counterclockwise centered at the origin

Explanation:

Step1: Analyze the size relationship

Trapezoid B is smaller than trapezoid A. So the scale factor should be less than 1. A dilation with scale factor \(2\) (options a and b) will make the figure larger, so we can eliminate options a and b.

Step2: Analyze the transformation after dilation

Let's assume a general point \((x,y)\) on trapezoid A. After a dilation with scale factor \(\frac{1}{2}\) centered at the origin, the point becomes \((\frac{1}{2}x,\frac{1}{2}y)\).
If we do a reflection across the \(y -\)axis, the transformation rule is \((x,y)\to(-x,y)\). If we do a \(90 -\)degree rotation counter - clockwise centered at the origin, the transformation rule is \((x,y)\to(-y,x)\).
Let's take a vertex of trapezoid A, say if we assume a vertex of trapezoid A is \((- 8,8)\). After dilation with scale factor \(\frac{1}{2}\), it becomes \((-4,4)\). After reflection across the \(y -\)axis, it becomes \((4,4)\) which is not the position of vertices of trapezoid B.
If we take a vertex of trapezoid A (say \((-8,8)\)), after dilation with scale factor \(\frac{1}{2}\) we get \((-4,4)\). After a \(90 -\)degree rotation counter - clockwise centered at the origin \((x,y)\to(-y,x)\), so \((-4,4)\to(-4,-4)\) (wrong). But if we consider the correct vertices (by visual inspection of the trapezoid on the coordinate - plane, assume a non - parallel side of trapezoid A. If we use the property of similarity: dilation (to change the size) and rotation (to match the orientation).
Let's use the rule of transformation. Suppose we have a trapezoid A. If we first dilate trapezoid A by a scale factor of \(\frac{1}{2}\) centered at the origin. The transformation for a point \((x,y)\) is \((x,y)\to(\frac{1}{2}x,\frac{1}{2}y)\). Then a \(90 -\)degree rotation counter - clockwise centered at the origin: \((x,y)\to(-y,x)\) will map trapezoid A to trapezoid B.

Answer:

D. a dilation by scale factor \(\frac{1}{2}\) centered at the origin followed by a \(90 -\)degree rotation counterclockwise centered at the origin