Question

Which of these triangle pairs can be mapped to each other using both a translation and a rotation about C?

Triangles X Y C and A B C are shown. Both triangles are congruent and share common point C. Triangle A B C is slightly lower than triangle X Y C.
Triangles X Y Z and A B C are shown. Both triangles are congruent. Triangle X Y Z is identical to triangle A B C but is slightly higher.
Triangles X Y Z and A B C are shown. Both triangles are congruent. Triangle X Y Z is reflected across a line to form triangle A B C.
Triangles X Y Z and A B C are shown. Both triangles are congruent. Triangle X Y Z is rotated down and to the left to form triangle A B C. It is also slightly higher than triangle A B C.

Answer 1

Only the fourth pair.

Why:
- Pair 1 (XYC ↔ ABC): they share C, so a rotation about C can move X,Y onto A,B, but a nontrivial translation cannot because it would move C (so only the identity translation would keep C fixed).
- Pair 2 (XYZ ↔ ABC): they are the same orientation and just shifted, so a translation works but a rotation about C (nontrivial) will not.
- Pair 3: one is the mirror image of the other (a reflection), so neither a pure rotation about C nor a translation can produce it.
- Pair 4: one triangle is both rotated and shifted relative to the other, so it can be obtained by a rotation (about an appropriate point, here C) together with a translation. Hence only the fourth pair can be mapped by both a translation and a rotation about C.