Question

describe the sequence of transformations that could be used to map \\( \triangle abc \\) onto \\( \triangle zyx \\).
a rotation of \\( 90 ^ { \circ } \\) clockwise about the origin, followed by a translation of \\( ( x, y ) \
ightarrow ( x - 1, y - 4 ) \\)
b rotation of \\( 180 ^ { \circ } \\) clockwise about the origin, followed by a translation of \\( ( x, y ) \
ightarrow ( x + 1, y + 2 ) \\)
c reflection in the line \\( y = x \\), followed by a translation of \\( ( x, y ) \
ightarrow ( x - 1, y ) \\)
d reflection in the \\( x \\)-axis, followed by a translation of \\( ( x, y ) \
ightarrow ( x + 5, y + 2 ) \\)

Explanation:

Step1: Analyze each option

• Option A:
• A rotation of \(90^{\circ}\) clockwise about the origin has the transformation rule \((x,y)\to(y, - x)\). Then a translation \((x,y)\to(x - 1,y - 4)\). This will not map \(\triangle ABC\) onto \(\triangle ZYX\) as the orientation and position changes do not match the required congruent mapping.
• Option B:
• A rotation of \(180^{\circ}\) clockwise about the origin has the transformation rule \((x,y)\to(-x,-y)\). Then a translation \((x,y)\to(x + 1,y+2)\). This will not map \(\triangle ABC\) onto \(\triangle ZYX\) as the orientation and position changes do not match the required congruent mapping.
• Option C:
• A reflection in the line \(y = x\) has the transformation rule \((x,y)\to(y,x)\). Then a translation \((x,y)\to(x - 1,y)\). This will not map \(\triangle ABC\) onto \(\triangle ZYX\) as the orientation and position changes do not match the required congruent mapping.
• Option D:
• A reflection in the \(x -\)axis has the transformation rule \((x,y)\to(x,-y)\). Then a translation \((x,y)\to(x + 5,y+2)\). This can map \(\triangle ABC\) onto \(\triangle ZYX\) as reflection preserves shape and size and the translation moves the reflected triangle to the position of \(\triangle ZYX\).

Answer:

D. Reflection in the x - axis, followed by a translation of \((x,y)\to(x + 5,y + 2)\)