Question

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

Explanation:

Step1: Analyze the first triangle pair (△XYZ and △ABC with vertex C)

• Check if translation and rotation about C work. Both triangles are right - angled, and sides are marked congruent. A translation can move one triangle to align with the other's position relative to C, and a rotation about C (since they share vertex C) can map the corresponding sides/angles. The triangles have congruent sides (marked with tick marks) and right angles, so translation to adjust position and rotation about C (as they are connected at C) can map them.

Step2: Analyze the second triangle pair (△XYZ and △ABC with A and C)

• These triangles are not connected at a common vertex for rotation about C in a way that translation and rotation would work. The first pair (with △XYZ and △ABC sharing C) is more suitable as they have a common vertex C, congruent sides, and right angles, allowing translation (to move the non - C parts into position) and rotation about C (to align the angles/sides).

Answer:

The triangle pair with triangles \( \triangle XYC \) (or \( \triangle XYZ \) with right angle at Y) and \( \triangle ABC \) (right angle at B) sharing vertex \( C \) (the first pair in the diagram) can be mapped to each other using both a translation and a rotation about \( C \).