Question

△def ~ △abc. what is the sequence of transformations that maps △abc to △def? a rotation of 90° about the origin and a translation 2 units down a reflection across the y - axis and a translation 2 units down a rotation of 180° about the origin and a dilation with center (0, 0) and scale factor 2 a reflection across the y - axis and a translation 4 units down

Explanation:

Step1: Analyze rotation/reflection

First, check rotation/reflection. A 180° rotation about origin: \((x,y)\to(-x,-y)\). A reflection over y - axis: \((x,y)\to(-x,y)\). Let's assume coordinates (though not fully visible, the key is to check the transformation sequence). The correct sequence should involve reflection over y - axis (to flip horizontally) and then translation down. The purple option says "A reflection across the y - axis and a translation 2 units down". Let's verify: Reflecting over y - axis changes the x - sign, then moving 2 units down (subtract 2 from y - coordinate) would map \(\triangle ABC\) to \(\triangle DEF\). Other options: 90° rotation (wrong direction), 180° rotation + dilation (dilation would change size, but triangles seem same size), 4 units down (wrong translation). So the purple option is correct.

Step2: Confirm the transformation

The reflection over y - axis (\((x,y)\to(-x,y)\)) and then translation 2 units down (\((x,y)\to(x,y - 2)\)) matches the mapping from \(\triangle ABC\) to \(\triangle DEF\) (visually, the horizontal flip and vertical shift down by 2).

Answer:

A reflection across the y - axis and a translation 2 units down (the purple - colored option)