Question

which sequences of transformations map fgh onto fgh? select all that apply. a translation left 4 units and up 2 units followed by a rotation 90° counterclockwise around the origin a reflection across the y - axis followed by a rotation 90° counterclockwise around the origin a rotation 180° around the origin followed by a translation right 1 unit and up 5 units

Explanation:

Step1: Recall transformation rules

Translation moves a figure, rotation turns it around a point, and reflection flips it over a line.

Step2: Analyze option 1

Translation left 4 units and up 2 units: $(x,y)\to(x - 4,y + 2)$. Then rotation 90° counter - clockwise around the origin: $(x,y)\to(-y,x)$. This sequence does not map $\triangle FGH$ onto $\triangle F'G'H'$.

Step3: Analyze option 2

Reflection across the y - axis: $(x,y)\to(-x,y)$. Then rotation 90° counter - clockwise around the origin: $(x,y)\to(-y,x)$. Substituting the reflection result into the rotation, we get $(x,y)\to(-y,-x)$. This sequence does not map $\triangle FGH$ onto $\triangle F'G'H'$.

Step4: Analyze option 3

Rotation 180° around the origin: $(x,y)\to(-x,-y)$. Then translation right 1 unit and up 5 units: $(x,y)\to(x + 1,y + 5)$. Let's assume a point $(x,y)$ on $\triangle FGH$. After rotation 180° it becomes $(-x,-y)$, and after translation it becomes $(-x + 1,-y+5)$. This sequence maps $\triangle FGH$ onto $\triangle F'G'H'$.

Answer:

a rotation 180° around the origin followed by a translation right 1 unit and up 5 units