Question

which of the following transformations maps uvw onto uvw? translation right 2 units and up 10 units; translation right 2 units and up 12 units; reflection across the x-axis; reflection across the y-axis; rotation 90° clockwise around the origin; rotation 90° counterclockwise around the origin; rotation 180° around the origin

Explanation:

Step1: Identify coordinates of U, V, W

Let's find coordinates: \( U(-7, -7) \), \( V(0, -9) \), \( W(-7, -9) \); \( U'(-5, 5) \), \( V'(2, 3) \), \( W'(-5, 3) \).

Step2: Analyze reflection across x - axis

Reflection across x - axis: \( (x,y)\to(x, - y) \). For \( U(-7,-7) \), it would be \( (-7,7) \), not \( U'(-5,5) \). So not reflection over x - axis.

Step3: Analyze reflection across y - axis

Reflection across y - axis: \( (x,y)\to(-x,y) \). For \( U(-7,-7) \), it would be \( (7,-7) \), not \( U'(-5,5) \). So not reflection over y - axis.

Step4: Analyze rotation 90° clockwise

Rotation 90° clockwise: \( (x,y)\to(y, - x) \). For \( U(-7,-7) \), it would be \( (-7,7) \), not \( U'(-5,5) \).

Step5: Analyze rotation 90° counter - clockwise

Rotation 90° counter - clockwise: \( (x,y)\to(-y,x) \). For \( U(-7,-7) \), it would be \( (7,-7) \), not \( U'(-5,5) \).

Step6: Analyze rotation 180°

Rotation 180°: \( (x,y)\to(-x,-y) \). For \( U(-7,-7) \), \( (-(-7),-(-7))=(7,7) \), not \( U'(-5,5) \).

Step7: Analyze translation

Check translation: Let's take point \( U(-7,-7) \) to \( U'(-5,5) \). Change in x: \( - 5-(-7)=2 \) (right 2 units). Change in y: \( 5 - (-7)=12 \) (up 12 units). Check \( V(0,-9) \) to \( V'(2,3) \): \( 2 - 0 = 2 \) (right 2), \( 3-(-9)=12 \) (up 12). Check \( W(-7,-9) \) to \( W'(-5,3) \): \( - 5-(-7)=2 \) (right 2), \( 3-(-9)=12 \) (up 12). So translation right 2 units and up 12 units works.

Answer:

translation right 2 units and up 12 units