Question

the graph shows triangles fgh and fgh. which sequence of transformations maps fgh onto fgh? a rotation 90° counterclockwise around the origin followed by a reflection across the y - axis a rotation 180° around the origin followed by a translation right 3 units and up 3 units a rotation 90° clockwise around the origin followed by a reflection across the y - axis

Explanation:

Step1: Analyze rotation rules

The general rule for a 90 - degree counter - clockwise rotation around the origin is $(x,y)\to(-y,x)$, for a 180 - degree rotation around the origin is $(x,y)\to(-x,-y)$ and for a 90 - degree clockwise rotation around the origin is $(x,y)\to(y, - x)$.

Step2: Analyze reflection rules

The rule for a reflection across the y - axis is $(x,y)\to(-x,y)$.

Step3: Check option 1

For a 90 - degree counter - clockwise rotation of a point $(x,y)$ on $\triangle FGH$ around the origin, and then a reflection across the y - axis, the combined transformation does not map $\triangle FGH$ onto $\triangle F'G'H'$.

Step4: Check option 2

For a 180 - degree rotation of a point $(x,y)$ on $\triangle FGH$ around the origin, we get $(-x,-y)$. Then a translation right 3 units and up 3 units gives $(-x + 3,-y+3)$. This sequence of transformations maps $\triangle FGH$ onto $\triangle F'G'H'$.

Step5: Check option 3

For a 90 - degree clockwise rotation of a point $(x,y)$ on $\triangle FGH$ around the origin, and then a reflection across the y - axis, the combined transformation does not map $\triangle FGH$ onto $\triangle F'G'H'$.

Answer:

a rotation 180° around the origin followed by a translation right 3 units and up 3 units