Question

which sequence of transformations maps vwxyz onto vwxyz? a rotation 90° clockwise around the origin followed by a reflection across the line y = 6 a rotation 180° around the origin followed by a reflection across the line x = 2 a rotation 90° counterclockwise followed by a reflection across the line y = x

Explanation:

Step1: Recall rotation rules

For a 90 - degree clockwise rotation around the origin, the transformation rule for a point $(x,y)$ is $(y, - x)$. For a 180 - degree rotation around the origin, the rule for a point $(x,y)$ is $(-x,-y)$. For a 90 - degree counter - clockwise rotation around the origin, the rule for a point $(x,y)$ is $(-y,x)$.

Step2: Analyze reflection rules

Reflection across the line $y = k$ for a point $(x,y)$ gives $(x,2k - y)$. Reflection across the line $x = h$ for a point $(x,y)$ gives $(2h - x,y)$. Reflection across the line $y=x$ for a point $(x,y)$ gives $(y,x)$.

Step3: Check each option

• Option 1: A 90 - degree clockwise rotation around the origin: $(x,y)\to(y, - x)$. Then reflection across $y = 6$: $(y,-x)\to(y,12 + x)$.
• Option 2: A 180 - degree rotation around the origin: $(x,y)\to(-x,-y)$. Then reflection across $x = 2$: $(-x,-y)\to(4 + x,-y)$.
• Option 3: A 90 - degree counter - clockwise rotation around the origin: $(x,y)\to(-y,x)$. Then reflection across $y=x$: $(-y,x)\to(x,-y)$.

Let's assume a general point $(x,y)$ on $VWXYZ$. After a 90 - degree clockwise rotation around the origin, $(x,y)$ becomes $(y,-x)$. Then, when we reflect $(y,-x)$ across the line $y = 6$, the $x$ - coordinate remains the same and the $y$ - coordinate is $2\times6-(-x)=12 + x$.
We can take a specific point on $VWXYZ$, say $(x = 1,y=-2)$. After 90 - degree clockwise rotation around the origin, it becomes $(-2,-1)$. After reflection across $y = 6$, the $x$ - coordinate is $-2$ and the $y$ - coordinate is $2\times6-(-1)=13$.
If we consider a 180 - degree rotation around the origin for the point $(1,-2)$, it becomes $(-1,2)$. Then reflection across $x = 2$ gives $(2\times2-(-1),2)=(5,2)$.
For a 90 - degree counter - clockwise rotation of $(1,-2)$ around the origin, it becomes $(2,1)$. Then reflection across $y = x$ gives $(1,2)$.
By observing the orientation and position of the pre - image $VWXYZ$ and the image $V'W'X'Y'Z'$, we can use the property of transformation of a single point. Let's take point $X(1,-2)$.
A 90 - degree clockwise rotation around the origin: $(1,-2)\to(-2,-1)$. Reflection across $y = 6$: $(-2,-1)\to(-2,13)$.
A 180 - degree rotation around the origin: $(1,-2)\to(-1,2)$. Reflection across $x = 2$: $(-1,2)\to(5,2)$.
A 90 - degree counter - clockwise rotation around the origin: $(1,-2)\to(2,1)$. Reflection across $y = x$: $(2,1)\to(1,2)$.
By checking the orientation of the polygons, a 90 - degree clockwise rotation around the origin followed by a reflection across the line $y = 6$ maps $VWXYZ$ onto $V'W'X'Y'Z'$.

Answer:

a rotation 90° clockwise around the origin followed by a reflection across the line y = 6