Question

1. in the diagram below, \\( \triangle abc \sim \triangle def \\).

which one of the following sequences of transformations maps \\( \triangle abc \\) onto \\( \triangle def \\)?
a) a rotation of \\( 180^{\circ} \\) about the origin followed by a translation
b) a counterclockwise rotation of \\( 90^{\circ} \\) about the origin followed by a translation
c) a reflection over the \\( x \\)-axis followed by a translation
d) a reflection over the \\( y \\)-axis followed by a translation

Explanation:

Step1: Analyze reflection over x - axis

When a point \((x,y)\) is reflected over the \(x -\)axis, it becomes \((x,-y)\). If we reflect \(\triangle ABC\) over the \(x -\)axis, the orientation of the triangle (in terms of up - down) will be reversed. But \(\triangle ABC\) and \(\triangle DEF\) have a different left - right orientation (not just up - down if we consider reflection over \(x -\)axis).

Step2: Analyze rotation of \(180^{\circ}\)

When a point \((x,y)\) is rotated \(180^{\circ}\) about the origin, it becomes \((-x,-y)\). A \(180^{\circ}\) rotation changes both the \(x\) and \(y\) coordinates' signs. But \(\triangle ABC\) and \(\triangle DEF\) do not have the relationship of a \(180^{\circ}\) rotation (if we assume a general position of non - symmetric - about - origin triangles).

Step3: Analyze counter - clockwise rotation of \(90^{\circ}\)

When a point \((x,y)\) is rotated counter - clockwise \(90^{\circ}\) about the origin, it becomes \((-y,x)\). This changes the orientation in a way that is not consistent with the given \(\triangle ABC\) and \(\triangle DEF\) (the side - length relationships and vertex - position relationships do not match a \(90^{\circ}\) counter - clockwise rotation result).

Step4: Analyze reflection over y - axis

When a point \((x,y)\) is reflected over the \(y -\)axis, it becomes \((-x,y)\). Reflecting \(\triangle ABC\) over the \(y -\)axis will change the left - right orientation. After reflection over the \(y -\)axis, a translation (shifting) is needed to map the reflected triangle onto \(\triangle DEF\) as the position (not just orientation) is also different.

Answer:

D. a reflection over the y - axis followed by a translation