Question

find the composition of transformations that map abcd to abcd. rotate clockwise about the origin ?°, then reflect over the -axis.

Explanation:

Step1: Determine the rotation angle

• Original rectangle \(ABCD\) has vertices in the second - quadrant. After rotation, we consider the orientation change.
• A \(90^{\circ}\) clock - wise rotation about the origin \((x,y)\to(y, - x)\). If we assume a general point \((-a,b)\) (for a rectangle in the second quadrant), after a \(90^{\circ}\) clock - wise rotation, it becomes \((b,a)\). But if we consider the relative position of the sides of the rectangle, a \(90^{\circ}\) clock - wise rotation about the origin changes the orientation of the rectangle.
• Let's check the direction of the sides. The vertical sides of \(ABCD\) (parallel to the \(y\) - axis) become horizontal sides (parallel to the \(x\) - axis) after a \(90^{\circ}\) clock - wise rotation.

Step2: Determine the reflection axis

• After a \(90^{\circ}\) clock - wise rotation about the origin, if we consider the position of the rotated rectangle (before reflection) and the final position of \(A'B'C'D'\).
• The transformation rule for reflection over the \(x\) - axis is \((x,y)\to(x, - y)\). If we assume a point \((x,y)\) after rotation, reflecting it over the \(x\) - axis will map it to the correct position of \(A'B'C'D'\)

Answer:

Rotate clockwise about the origin \(90^{\circ}\), then reflect over the \(x\) - axis.