Question

which sequence of transformations maps opq onto opq? a rotation 180° around the origin followed by a reflection across the line y = 1 a translation up 13 units followed by a rotation 180° around the origin a reflection across the line y = -5 followed by a translation left 9 units

Explanation:

Step1: Recall rotation rules

A $180^{\circ}$ rotation around the origin for a point $(x,y)$ gives $(-x,-y)$.

Step2: Analyze option 1

If we first rotate $\triangle OPQ$ $180^{\circ}$ around the origin, the orientation and position of the triangle will change. Then reflecting across $y = 1$ will further change its position. But this is not the correct sequence.

Step3: Analyze option 2

If we first translate $\triangle OPQ$ up 13 units, the $y$-coordinates of all points will increase by 13. Then rotating $180^{\circ}$ around the origin will not map it to $\triangle O'P'Q'$.

Step4: Analyze option 3

Let's consider the reflection across $y=-5$. The distance between a point $(x,y)$ and the line $y = - 5$ is $d=y+5$. After reflection across $y=-5$, the new $y$-coordinate is $y'=-5-(y + 5)=-y - 10$. Then translating left 9 units (subtracting 9 from the $x$-coordinate) will map $\triangle OPQ$ onto $\triangle O'P'Q'$.

Answer:

a reflection across the line $y=-5$ followed by a translation left 9 units