Question

which sequence of transformations maps tuv onto tuv? a translation right 12 units followed by a reflection across the x - axis a rotation 90° counterclockwise around the origin followed by a translation up 13 units a reflection across the x - axis followed by a rotation 90° clockwise around the origin

Explanation:

Step1: Analyze translation right 12 units and x - axis reflection

Let's assume a general point \((x,y)\) on \(\triangle TUV\). A translation right 12 units gives \((x + 12,y)\), and a reflection across the \(x\) - axis gives \((x + 12,-y)\). If we consider the coordinates of the vertices of \(\triangle TUV\) and apply these transformations, we can see it does not map to \(\triangle T'U'V'\).

Step2: Analyze 90 - degree counter - clockwise rotation and translation up 13 units

The rule for a 90 - degree counter - clockwise rotation around the origin is \((x,y)\to(-y,x)\). Let the vertices of \(\triangle TUV\) be \((x_1,y_1),(x_2,y_2),(x_3,y_3)\). After rotation, they become \((-y_1,x_1),(-y_2,x_2),(-y_3,x_3)\). Then a translation up 13 units gives \((-y_1,x_1 + 13),(-y_2,x_2+13),(-y_3,x_3 + 13)\). By checking the coordinates of the vertices of \(\triangle TUV\) and \(\triangle T'U'V'\), we find this sequence of transformations maps \(\triangle TUV\) onto \(\triangle T'U'V'\).

Step3: Analyze x - axis reflection and 90 - degree clockwise rotation

A reflection across the \(x\) - axis takes \((x,y)\) to \((x,-y)\). A 90 - degree clockwise rotation around the origin takes \((x,y)\) to \((y,-x)\). If we first reflect \((x,y)\) across the \(x\) - axis to get \((x,-y)\) and then rotate 90 - degree clockwise, we get \((-y,-x)\). This does not map \(\triangle TUV\) onto \(\triangle T'U'V'\).

Answer:

a rotation 90° counterclockwise around the origin followed by a translation up 13 units