Triangle ABC is rotated to create the image A'B'C'.

Question

On a coordinate plane, 2 triangles are shown. The first triangle has points A (negative 1, negative 1), B (1, negative 1), and C (0, negative 4). The second triangle has points A prime (1, 1), B prime (negative 1, 1), and C prime (0, 4).
Which rule describes the transformation?

(x, y) → (x, –y)
(x, y) → (y, x)
(x, y) → (–x, –y)
(x, y) → (–y, –x)

Answer 1

To determine the transformation rule that describes how triangle ABC is rotated to create triangle A'B'C', we can compare the coordinates of the points in both triangles.

The points of triangle ABC are:

• A(-1, -1)
• B(1, -1)
• C(0, -4)

The points of triangle A'B'C' are:

• A'(1, 1)
• B'(-1, 1)
• C'(0, 4)

Now, let's analyze how each point is transformed:

1. For point A(-1, -1) to A'(1, 1):
   • The x-coordinate changes from -1 to 1: this corresponds to (x → -x).
   • The y-coordinate changes from -1 to 1: this corresponds to (y → -y).

So, A transforms according to the rule: (x, y) → (-x, -y).

2. For point B(1, -1) to B'(-1, 1):
   • The x-coordinate changes from 1 to -1: this corresponds to (x → -x).
   • The y-coordinate changes from -1 to 1: this corresponds to (y → -y).

So, B transforms according to the rule: (x, y) → (-x, -y).

3. For point C(0, -4) to C'(0, 4):
   • The x-coordinate remains 0: unchanged.
   • The y-coordinate changes from -4 to 4: this corresponds to (y → -y).

So, C also transforms according to the rule: (x, y) → (x, -y). However, for this point, since x doesn't change, it does not affect the transformation entirely.

From all points analyzed, the overall transformation rule resembles that of reflecting over the x-axis, which can be expressed as:

\[(x, y) \rightarrow (-x, -y)\]

Therefore, the correct answer is:

\((x, y) \rightarrow (-x, -y)\)