Question

the graph shows pentagons ghijk and ghijk. which sequences of transformations map ghijk onto ghijk? select all that apply. a rotation 90° clockwise around the origin followed by a reflection across the y - axis a translation right 8 units and down 1 unit 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 90° counterclockwise around the origin followed by a translation right 1 unit and up 8 units

Explanation:

Step1: Recall rotation rules

A $90^{\circ}$ clock - wise rotation around the origin has the rule $(x,y)\to(y, - x)$. A reflection across the $y$ - axis has the rule $(x,y)\to(-x,y)$. Analyze each option.

Step2: Analyze option 1

A rotation $90^{\circ}$ clockwise around the origin followed by a reflection across the $y$ - axis:
Let the original point be $(x,y)$. After a $90^{\circ}$ clockwise rotation around the origin, it becomes $(y,-x)$. After a reflection across the $y$ - axis, it becomes $(-y,-x)$.

Step3: Analyze option 2

A translation right 8 units and down 1 unit followed by a rotation $90^{\circ}$ counter - clockwise around the origin:
A translation right 8 units and down 1 unit has the rule $(x,y)\to(x + 8,y-1)$. A $90^{\circ}$ counter - clockwise rotation around the origin has the rule $(x,y)\to(-y,x)$.

Step4: Analyze option 3

A reflection across the $y$ - axis followed by a rotation $90^{\circ}$ counter - clockwise around the origin:
A reflection across the $y$ - axis: $(x,y)\to(-x,y)$. A $90^{\circ}$ counter - clockwise rotation around the origin: $(-x,y)\to(-y,-x)$.

Step5: Analyze option 4

A rotation $90^{\circ}$ counter - clockwise around the origin followed by a translation right 1 unit and up 8 units:
A $90^{\circ}$ counter - clockwise rotation around the origin: $(x,y)\to(-y,x)$. A translation right 1 unit and up 8 units: $(-y,x)\to(-y + 1,x + 8)$.

By comparing the coordinates of the vertices of the original pentagon $GHIJK$ and the transformed pentagon $G'H'I'J'K'$ through coordinate - geometry of transformations, we find that the correct sequences are:
A reflection across the $y$ - axis followed by a rotation $90^{\circ}$ counter - clockwise around the origin.
A rotation $90^{\circ}$ counter - clockwise around the origin followed by a translation right 1 unit and up 8 units.

Answer:

The correct sequences of transformations are:

• A reflection across the $y$ - axis followed by a rotation $90^{\circ}$ counter - clockwise around the origin.
• A rotation $90^{\circ}$ counter - clockwise around the origin followed by a translation right 1 unit and up 8 units.