Question

Which statement explains why △ABC is congruent to △A′B′C′
?

Responses

You can map △ABC onto △A′B′C′
by rotating it 180° about the origin and translating it 2 units right, which is a sequence of rigid motions.
You can map triangle A B C onto △ A ′ B ′ C ′ by rotating it 180° about the origin and translating it 2 units right, which is a sequence of rigid motions.

You can map △ABC onto △A′B′C′
by reflecting it across the x-axis and translating it 6 units left, which is a sequence of rigid motions.
You can map triangle A B C onto △ A ′ B ′ C ′ by reflecting it across the x -axis and translating it 6 units left, which is a sequence of rigid motions.

You can map △ABC onto △A′B′C′
by reflecting it across the y-axis and translating it 6 units down, which is a sequence of rigid motions.
You can map triangle A B C onto △ A ′ B ′ C ′ by reflecting it across the y -axis and translating it 6 units down, which is a sequence of rigid motions.

You can map △ABC onto △A′B′C′
by rotating it 90°clockwise about the origin and translating it 4 units left, which is a sequence of rigid motions.
You can map triangle A B C onto △ A ′ B ′ C ′ by rotating it 90°clockwise about the origin and translating it 4 units left, which is a sequence of rigid motions.
Two triangles on the coordinate plane. Triangle A B C has vertex A at 4 comma 5, vertex B at 1 comma 1, and vertex C at 5 comma negative 1. Triangle A prime B prime C prime has vertex A prime at negative 2 comma negative 5, vertex B prime at 1 comma negative 1, and vertex C prime at negative 3 comma 1.

Answer 1

The first statement. A 180° rotation about the origin sends (x,y) to (−x,−y), so A(4,5)→(−4,−5), B(1,1)→(−1,−1), C(5,−1)→(−5,1). Translating 2 units right (add (2,0)) gives A′(−2,−5), B′(1,−1), C′(−3,1). A composition of rigid motions preserves distance, so △ABC ≅ △A′B′C′.