Question

Which of these triangle pairs can be mapped to each other using two reflections?

Triangles X Y Z and P Y T are connected at point Y. Triangle X Y Z is reflected across X Y and then is reflected another time across Y Z to form triangle P Y T.
Triangles X Y Z and A Y T are connected at point Y. Triangle X Y Z is reflected 3 times to form triangle A Y T.
Triangles X Y Z and A Y T are connected at point Y. Triangle X Y Z is reflected across side X Y and then is shifted down to form triangle A Y T.
Triangles X Y Z and A Z T are connected at point Z. Triangle X Y Z is reflected and then rotated to form triangle A Z T.

Answer 1

Only the first pair (Triangles X Y Z and P Y T).

Reasoning (brief):
- A composition of two reflections is always orientation-preserving: it is either a rotation (if the reflection lines intersect) or a translation (if they are parallel).
- Case 1: reflecting across XY then across YZ (lines intersect at Y) is a rotation about Y — this is exactly two reflections, so XYZ can map to PYT. ⇒ Yes.
- Case 2: three reflections is an odd number of reflections (orientation-reversing) — cannot be realized by only two reflections. ⇒ No.
- Case 3: a reflection followed by a translation is orientation-reversing (a glide/reflection type), so it cannot equal two reflections (which are orientation-preserving). ⇒ No.
- Case 4: a reflection followed by a rotation is orientation-reversing (odd total), so it cannot be realized by two reflections. ⇒ No.