Is ΔP’Q’R’ a 180° counterclockwise rotation about the origin of ΔPQR? Explain. The coordinates of the image and preimage opposites.

To determine if ΔP’Q’R’ is a 180° counterclockwise rotation about the origin of ΔPQR, we can analyze the coordinates of corresponding points before and after the rotation.

When a point ( (x, y) ) is rotated 180° counterclockwise around the origin, its new coordinates become ( (-x, -y) ). This means we take each coordinate of the original triangle and switch its signs.

1. Identify the coordinates:

– Let the coordinates of points in ΔPQR be ( P(x_1, y_1) ), ( Q(x_2, y_2) ), and ( R(x_3, y_3) ).

– Then the coordinates after the 180° rotation would be ( P'(-x_1, -y_1) ), ( Q'(-x_2, -y_2) ), and ( R'(-x_3, -y_3) ).

2. Check the relationship:

– If the coordinates of ΔP’Q’R’ match the points ( P'(-x_1, -y_1) ), ( Q'(-x_2, -y_2) ), and ( R'(-x_3, -y_3) ), then ΔP’Q’R’ is indeed a 180° counterclockwise rotation of ΔPQR