Question

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)Responses90 degrees (counterclockwise)90 degrees (counterclockwise)−90 degrees (clockwise)negative 90 degrees (clockwise)270 degrees (counterclockwise)270 degrees (counterclockwise)−180 degrees (clockwise) or 180 degrees (counterclockwise)

Answer 1

To determine the degree and direction of rotation from point $P(-4,10)$ to point $P^{\prime }(-10,-4)$, we can apply the rotation mapping.

For a point $(x,y)$, the standard rotation mappings about the origin are:

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

Now let’s analyze the transformation that takes $P(-4,10)$ to $P^{\prime }(-10,-4)$:

1. Starting point $P(-4,10)$:

   • After 90 degrees counterclockwise: $$(-10,-4)(thisis{P}^{\prime })$$ Thus, applying the mapping confirms:

   $$P(-4,10)\to (-10,-4)\text{ after a rotation of }90\text{ degrees counterclockwise.}$$

Since the transformation from $P$ to $P^{\prime }$ matches the 90 degrees counterclockwise mapping, the answer is:

90 degrees (counterclockwise).