To rotate point $$P(7,3)$$ by $$90°$$ counterclockwise about $$T(3,1)$$: First, translate so $$T$$ is at origin: $$P_{temp} = (7-3, 3-1) = (4,2)$$. Then rotate $$90°$$ counterclockwise: $$(x,y) → (-y,x)$$, so $$(4,2) → (-2,4)$$. Finally, translate back: $$P' = (-2+3, 4+1) = (1,5)$$. Choice B uses clockwise rotation. Choice C forgets the final translation back. Choice D uses $$180°$$ rotation instead of $$90°$$.

After reflecting $$\triangle ABC$$ across the $$y$$-axis, we get $$\triangle A'B'C'$$ which is congruent to the original but with opposite orientation (if we consider the order of vertices). To map this onto $$\triangle DEF$$, we need one more rigid transformation. This could be: (1) a translation if $$\triangle A'B'C'$$ and $$\triangle DEF$$ have the same orientation and size, (2) a rotation if they have different orientations, or (3) a combination rotation and translation (which can be achieved as a single rotation about an appropriate point). The key insight is that any rigid transformation preserves distance and angles, so any single transformation that aligns the triangles will work. Choice A is too restrictive. Choice B assumes rotation about origin is required. Choice D would create a specific orientation that may not match $$\triangle DEF$$.

This is a sequencing rigid transformations problem where we need to map triangle RST onto triangle R'S'T'. The transformations needed are a rotation and a translation. The order matters because rotations about the origin affect both position and orientation. The correct sequence is to rotate 90° counterclockwise about the origin first, then translate down 2 units. This works because the rotation reorients the triangle and moves it to a new position around the origin, then the translation shifts it down to reach the final location. If we translated down first then rotated (choice B), the lowered triangle would follow a circular path during rotation and end up in the wrong position. The key principle is to perform rotations about the origin before translations to ensure the correct final placement.

This is a sequencing rigid transformations problem where we need to map triangle DEF onto triangle D'E'F'. The transformations needed are a rotation and a translation. The order matters significantly because rotations about the origin change both orientation and position. The correct sequence is to rotate 90° clockwise about the origin first, then translate up 3 units. This works because the rotation reorients the triangle and moves it to a new position relative to the origin, then the translation shifts it up to the final location. If we translated up first then rotated (choice B), the triangle would end up in the wrong position because rotation about the origin would move the already-elevated triangle in a circular path. The key principle is to perform rotations about the origin before translations to achieve the desired final position.