Question

the graph shows parallelogram lm nop and lmnop. which sequences of transformations map lm nop onto lmnop? select all that apply.
a a reflection across the x - axis followed by a rotation 90° counter - clockwise around the origin
b a translation right 7 units and up 1 unit followed by a rotation 90° clockwise around the origin
c a reflection across the y - axis followed by a rotation 90° clockwise around the origin
d a rotation 90° clockwise around the origin followed by a translation right 1 unit and down 7 units

Explanation:

Step1: Recall transformation rules

• Reflection across x - axis: $(x,y)\to(x, - y)$
• 90 - degree counter - clockwise rotation around origin: $(x,y)\to(-y,x)$
• Reflection across y - axis: $(x,y)\to(-x,y)$
• 90 - degree clockwise rotation around origin: $(x,y)\to(y,-x)$
• Translation: $(x,y)\to(x + a,y + b)$ where $a$ is horizontal and $b$ is vertical displacement.

Step2: Analyze each option

Option A

• Reflection across x - axis: $(x,y)\to(x, - y)$. Then 90 - degree counter - clockwise rotation around origin: $(x,-y)\to(y,x)$. This does not match the transformation from LM NOP to L'M'N'O'P'.

Option B

• Reflection across y - axis: $(x,y)\to(-x,y)$. Then 90 - degree clockwise rotation around origin: $(-x,y)\to(y,x)$. This does not match the transformation from LM NOP to L'M'N'O'P'.

Option C

• Translation right 7 units and up 1 unit: $(x,y)\to(x + 7,y + 1)$. Then 90 - degree clockwise rotation around origin: $(x + 7,y + 1)\to(y + 1,-(x + 7))$. This does not match the transformation from LM NOP to L'M'N'O'P'.

Option D

• 90 - degree clockwise rotation around origin: $(x,y)\to(y,-x)$. Then translation right 1 unit and down 7 units: $(y,-x)\to(y + 1,-x-7)$. This matches the transformation from LM NOP to L'M'N'O'P'.

Answer:

D. a rotation 90° clockwise around the origin followed by a translation right 1 unit and down 7 units