A point is rotated about the origin. Its coordinates are P(−4,10) and P′(−10,−4) . Determine the degree of rotation and direction by applying rotation mapping.(1 point) Responses 90 degrees (counterclockwise) 90 degrees (counterclockwise) 270 degrees (counterclockwise) 270 degrees (counterclockwise) −90 degrees (clockwise) negative 90 degrees (clockwise) −180 degrees (clockwise) or 180 degrees (counterclockwise)

Question

GPT-4o mini · Answer

To determine the degree of rotation and the direction, we can analyze the transformation of the point \( P(-4, 10) \) to \( P'(-10, -4) \).

A rotation about the origin can be characterized by the following coordinate transformations for a point \( (x, y) \):

90 degrees counterclockwise: \( (x, y) \to (-y, x) \)
180 degrees: \( (x, y) \to (-x, -y) \)
270 degrees counterclockwise (or equivalently, 90 degrees clockwise): \( (x, y) \to (y, -x) \)

Let's check the transformation needed to go from \( P(-4, 10) \) to \( P'(-10, -4) \):

Check 90 degrees counterclockwise: \[ (-4, 10) \to (-10, -4) \] Here, the transformation would give \( ( -y, x ) \to (-10, -4)\) which matches \( P' \). This indicates a 90 degrees counterclockwise rotation.

Now, we can conclude that the point \( P(-4, 10) \) is rotated to \( P'(-10, -4) \) by 90 degrees counterclockwise.

Thus, the answer is 90 degrees (counterclockwise).