Question

which is a way to show △def is congruent to △def? a reflection across the y - axis and a translation 1 unit up and 1 unit left a rotation 180° about the origin and a translation 1 unit up a reflection across the line y = 1 and a translation 2 units up a rotation 90° and a translation 1 unit up and 1 unit left

Explanation:

Step1: Recall congruence - transformation rules

Congruent triangles can be obtained by rigid - motions like reflections, rotations, and translations.

Step2: Analyze each option

• Option A: A reflection across the y - axis changes the sign of the x - coordinates of the vertices. A translation 1 unit up and 1 unit left will further change the position.
• Option B: A 180° rotation about the origin changes the signs of both x and y coordinates of the vertices. A translation 1 unit up will move the triangle vertically.
• Option C: A reflection across the line y = 1 changes the y - coordinate of each point with respect to the line y = 1. A translation 2 units up will further move the triangle vertically.
• Option D: A 90° rotation changes the orientation of the triangle. A translation 1 unit up and 1 unit left will move the triangle.

We can check the orientation and position of the vertices of \(\triangle DEF\) and \(\triangle D'E'F'\) on the graph. By observing the orientation and position of the triangles, we find that a 180° rotation about the origin and a translation 1 unit up will map \(\triangle DEF\) onto \(\triangle D'E'F'\).

Answer:

B. A rotation 180° about the origin and a translation 1 unit up