Question

which sequences of transformations map tuv onto tuv? select all that apply. a rotation 180° around the origin followed by a translation left 2 units and down 10 units a translation right 10 units and up 8 units followed by a reflection across the x - axis a reflection across the x - axis followed by a rotation 180° around the origin a rotation 90° counterclockwise around the origin followed by a translation right 1 unit and up 2 units

Explanation:

Step1: Recall transformation rules

For a point $(x,y)$ rotated 180° around the origin, the new - point is $(-x,-y)$. For a translation left $a$ units and down $b$ units, the transformation is $(x,y)\to(x - a,y - b)$. For a translation right $a$ units and up $b$ units, the transformation is $(x,y)\to(x + a,y + b)$. For a reflection across the $x$ - axis, the transformation is $(x,y)\to(x,-y)$. For a 90° counter - clockwise rotation around the origin, the transformation is $(x,y)\to(-y,x)$.

Step2: Analyze the first option

Let's assume a point $(x,y)$ on $\triangle TUV$. A 180° rotation around the origin gives $(-x,-y)$. Then a translation left 2 units and down 10 units gives $(-x - 2,-y - 10)$. By checking the coordinates of the vertices of $\triangle TUV$ and $\triangle T'U'V'$, this sequence does not map $\triangle TUV$ onto $\triangle T'U'V'$.

Step3: Analyze the second option

Let a point on $\triangle TUV$ be $(x,y)$. A translation right 10 units and up 8 units gives $(x + 10,y + 8)$. Then a reflection across the $x$ - axis gives $(x + 10,-(y + 8))$. By checking the coordinates of the vertices of $\triangle TUV$ and $\triangle T'U'V'$, this sequence maps $\triangle TUV$ onto $\triangle T'U'V'$.

Step4: Analyze the third option

Let a point on $\triangle TUV$ be $(x,y)$. A reflection across the $x$ - axis gives $(x,-y)$. Then a 180° rotation around the origin gives $(-x,y)$. By checking the coordinates of the vertices of $\triangle TUV$ and $\triangle T'U'V'$, this sequence does not map $\triangle TUV$ onto $\triangle T'U'V'$.

Step5: Analyze the fourth option

Let a point on $\triangle TUV$ be $(x,y)$. A 90° counter - clockwise rotation around the origin gives $(-y,x)$. Then a translation right 1 unit and up 2 units gives $(-y + 1,x + 2)$. By checking the coordinates of the vertices of $\triangle TUV$ and $\triangle T'U'V'$, this sequence does not map $\triangle TUV$ onto $\triangle T'U'V'$.

Answer:

a translation right 10 units and up 8 units followed by a reflection across the $x$-axis