Question

which congruence transforma
a. reflection in the y - axis, followed by a reflection in the x - axis
b. rotation of 90° counterclockwise about the origin, followed by a reflection in the y - axis
c. rotation of 270° counterclockwise about the origin, followed by a reflection in the x - axis
d. reflection in the x - axis, followed by a reflection in the y - axis
e. reflection in the x - axis, followed by a rotation of 90° counterclockwise about the origin

Explanation:

Step1: Analyze Option A

A reflection in the y - axis followed by a reflection in the x - axis. Let \((x,y)\) be a point. After reflection in the y - axis, the point becomes \((-x,y)\). After reflection in the x - axis, it becomes \((-x,-y)\). This is equivalent to a rotation of \(180^{\circ}\) about the origin. But looking at the graph, this is not the transformation.

Step2: Analyze Option B

A rotation of \(90^{\circ}\) counterclockwise about the origin followed by a reflection in the y - axis. Let \((x,y)\) be a point. After a \(90^{\circ}\) counter - clockwise rotation about the origin, the point becomes \((-y,x)\). After reflection in the y - axis, it becomes \((y,x)\). This does not match the transformation in the graph.

Step3: Analyze Option C

A rotation of \(270^{\circ}\) counterclockwise about the origin followed by a reflection in the x - axis. Let \((x,y)\) be a point. A \(270^{\circ}\) counter - clockwise rotation about the origin gives \((y,-x)\). After reflection in the x - axis, it becomes \((y,x)\). This does not match the transformation in the graph.

Step4: Analyze Option D

A reflection in the x - axis followed by a reflection in the y - axis. Let \((x,y)\) be a point. After reflection in the x - axis, the point is \((x,-y)\). After reflection in the y - axis, it is \((-x,-y)\). This is equivalent to a rotation of \(180^{\circ}\) about the origin. But looking at the graph, this is not the transformation.

Step5: Analyze Option E

A reflection in the x - axis followed by a rotation of \(90^{\circ}\) counterclockwise about the origin. Let \((x,y)\) be a point. After reflection in the x - axis, the point is \((x,-y)\). After a \(90^{\circ}\) counter - clockwise rotation about the origin (\((x,y)\to(-y,x)\)), the point \((x,-y)\) becomes \((y,x)\). If we consider the vertices of the triangles (by taking sample points from the graph), this transformation matches.

Answer:

E. reflection in the x - axis, followed by a rotation of \(90^{\circ}\) counterclockwise about the origin