Question

the triangles are congruent by the sss congruence theorem. which transformation(s) can map △bcd onto △wxy? rotation only reflection only translation, then rotation translation, then reflection

Explanation:

• Translation: First, we can move (translate) \(\triangle BCD\) so that one of its vertices aligns with a corresponding vertex of \(\triangle WXY\). For example, we can translate \(\triangle BCD\) so that point \(C\) (or another vertex) gets closer in position to the corresponding vertex in \(\triangle WXY\) in terms of the overall spatial layout.
• Reflection: After translation, we observe the orientation of the two triangles. Since reflection changes the orientation of a figure (flips it over a line), after translating \(\triangle BCD\) to a position where its sides are in a similar spatial relationship (in terms of length - because of SSS congruence) to \(\triangle WXY\), a reflection (over an appropriate line) can map \(\triangle BCD\) onto \(\triangle WXY\).
• Why other options are wrong:
• Rotation only: Rotation changes the orientation of a figure around a point. But just rotating \(\triangle BCD\) cannot account for the change in the "handedness" (orientation in terms of left - right or up - down flipping) needed to map it onto \(\triangle WXY\) as per their relative positions in the given diagram.
• Reflection only: Without translation, we cannot position \(\triangle BCD\) in the correct spatial location (in terms of proximity of vertices) before reflecting it to match \(\triangle WXY\).
• Translation then rotation: Rotation does not correct the "flipping" (change in orientation that is like a mirror - image) that is required to map \(\triangle BCD\) onto \(\triangle WXY\) as per their given relative shapes and positions.

Answer:

translation, then reflection