Question

determine a series of transformations that would map polygon abcde onto polygon abcde.

Explanation:

Step1: Analyze Rotation

First, we can rotate polygon \( ABCDE \) 90 degrees counterclockwise (or 270 degrees clockwise) about the origin. The general rule for a 90 - degree counterclockwise rotation about the origin \((x,y)\to(-y,x)\). This rotation will align the orientation of the polygon closer to \( A'B'C'D'E' \).

Step2: Analyze Translation

After rotation, we need to translate the polygon. Let's consider the coordinates. For example, if we take a vertex of \( ABCDE \), say \( B(-7, - 1)\). After a 90 - degree counterclockwise rotation, \( (x,y)=(-7,-1)\) becomes \( (y,-x)=( - 1,7)\) (wait, correction: the correct rule for 90 - degree counterclockwise rotation is \((x,y)\to(-y,x)\), so for \( B(-7,-1)\), it becomes \( (1, - 7)\)? No, maybe a better approach is to look at the final position. Alternatively, we can rotate 90 degrees clockwise (rule \((x,y)\to(y,-x)\)) and then translate. Let's assume we rotate 90 degrees clockwise first. For a point \((x,y)\) in \( ABCDE \), after 90 - degree clockwise rotation, it becomes \((y,-x)\). Then, we translate the rotated figure. Let's take vertex \( A(-5,-2)\). After 90 - degree clockwise rotation: \( (x,y)=(-5,-2)\to(-2,5)\). Now, the vertex \( A' \) is at \( (4,4)\). So the translation vector would be \((4 - (-2),4 - 5)=(6,-1)\)? Wait, maybe a simpler way: first, rotate the polygon \( ABCDE \) 90 degrees clockwise (or counterclockwise) to get the same orientation as \( A'B'C'D'E' \), then translate it horizontally and vertically to match the position. Another approach: the polygon \( ABCDE \) can be rotated 90 degrees counterclockwise (or clockwise) and then translated. Let's confirm the orientation. The pink polygon \( ABCDE \) and the blue \( A'B'C'D'E' \) have the same shape, so rotation is needed first. Let's say we rotate \( ABCDE \) 90 degrees counterclockwise about the origin. Then, we translate it. Let's take point \( D \) in \( ABCDE \): let's find its coordinates. From the graph, \( D \) in \( ABCDE \) seems to be around \((-6,-6)\)? Wait, no, the pink polygon: \( B(-7,-1)\), \( A(-5,-2)\), \( C(-9,-7)\), \( D(-7,-6)\), \( E(-4,-8)\)? Maybe. After 90 - degree counterclockwise rotation, \( (x,y)\to(-y,x)\). For \( D(-7,-6)\), it becomes \( (6,-7)\). The \( D' \) in \( A'B'C'D'E' \) is at \( (7,0)\)? Wait, maybe I made a mistake in coordinates. Alternatively, the correct series of transformations: First, rotate polygon \( ABCDE \) 90 degrees clockwise about the origin (transformation rule \((x,y)\to(y, - x)\)). Then, translate the rotated polygon 8 units to the right and 5 units up (or other translation vector). Let's check with vertex \( A(-5,-2)\): after 90 - degree clockwise rotation: \( (x,y)=(-5,-2)\to(-2,5)\). Then, translating 6 units to the right (\(-2 + 6 = 4\)) and \(-1\) unit down (\(5-1 = 4\)) gives \( (4,4)\), which is \( A' \). So the rotation (90 degrees clockwise) followed by a translation (6 units right, 1 unit down? Wait, \( 5-1 = 4\), so from \( y = 5\) to \( y = 4\), that's a translation of \(-1\) in \( y \)-direction. So the series of transformations is: Rotate the polygon \( ABCDE \) 90 degrees clockwise about the origin, then translate it 6 units to the right and 1 unit down (or other translation based on coordinates). Alternatively, rotate 90 degrees counterclockwise and then translate. Another way: the polygon can be rotated 90 degrees counterclockwise (or clockwise) and then translated. So a possible series is: Rotation (90 degrees clockwise about the origin) followed by Translation (e.g., 8 units right and 5 units up, but let's get the correct translation). Let's take vertex \( B(-7,-1)\): after 90 - degree clockwise rotation: \( (x,y)=(-7,-1)\to(-1,7)\). Now, \( B' \) is at \( (3,5)\)? Wait, no, \( B' \) is at \( (3,5)\)? From the graph, \( B' \) is at \( (3,5)\)? Wait, the blue polygon: \( A'(4,4)\), \( B'(3,5)\), \( D'(7,0)\), \( C'(9,-1)\), \( E'(2,-2)\). So for \( B(-7,-1)\), after 90 - degree clockwise rotation: \( (y,-x)=(-1,7)\). Then, to get to \( (3,5)\), we need to translate \( (-1 + 4,7 - 2)=(3,5)\). So translation vector is \( (4,-2)\). So the rotation (90 degrees clockwise) and then translation by \( (4,-2) \). So the series of transformations is: Rotate polygon \( ABCDE \) 90 degrees clockwise about the origin, then translate it 4 units to the right and 2 units down.

Answer:

One possible series of transformations is: Rotate polygon \( ABCDE \) 90 degrees clockwise about the origin, then translate it 4 units to the right and 2 units down (or other valid rotation - translation combination, e.g., rotate 90 degrees counterclockwise and translate appropriately).