Question

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

Explanation:

Step1: Analyze translation and rotation requirements

Translation moves a figure without rotation/reflection, rotation about \( C \) means \( C \) is a fixed center. So the triangles must be congruent, with \( C \) as a common vertex (for rotation) and translatable (same orientation/position relation after translation).

Step2: Check each pair

• First pair: Triangles \( ABC \) and \( XYZ \). \( C \) and \( Z \) are separate, rotation about \( C \) would keep \( C \) fixed, but \( Z \) is not \( C \), so no.
• Second pair: Triangles \( ABC \) and \( XYZ \). \( C \) and \( Z \) are separate, rotation about \( C \) can't map to \( Z \) as center.
• Third pair: Both triangles share vertex \( C \) (rotation center). They are congruent (right triangles, equal marked sides). After translation (to align \( C \) if needed) and rotation about \( C \), they can map.
• Fourth pair: \( C \) and \( Z \) are separate vertices, rotation about \( C \) can't involve \( Z \) as center.

Answer:

The third triangle pair (the one with both triangles sharing vertex \( C \))