Final answer:
The provided rotations do not verify the congruence of the triangles after a 90-degree rotation because the rule for a 90-degree clockwise rotation is (x, y) → (y, -x), and none of the choices match this. However, if considered a 180-degree rotation, the correct rule is (x, y) → (-x, -y), which would indicate that the triangles are congruent.
Step-by-step explanation:
To determine if triangles A(2,-1), B(4,1), C(3,3) and A'(-2,1), B'(-4,-1), C'(-3,-3) are congruent after a 90-degree rotation, we need to identify the correct rotation rule. When a point is rotated 90 degrees clockwise around the origin in a coordinate plane, the transformation rule can be summarized as (x, y) → (y, -x). This rule switches the x and y coordinates and negates the new x-coordinate (which was the original y). By applying this rule to triangle ABC, we can check if the resulting coordinates match those of triangle A'B'C'.
Let's apply this to point A(2,-1):
A(2, -1) → (-1, -2), which does not match A'(-2, 1). Therefore, the given triangles are not congruent after a 90-degree clockwise rotation. The correct answer is:
D) the triangles are not congruent
However, a rotation of 180 degrees is represented by the rule (x,y) → (-x,-y), which would indeed map triangle ABC to A'B'C', since applying this rule to each point of triangle ABC gives the points of A'B'C':
- A(2,-1) → (-2,1)
- B(4,1) → (-4,-1)
- C(3,3) → (-3,-3)