Question

\(P(-7,2)\) is rotated \(270\degree\) (counterclockwise) about the origin. Name the quadrant \(P'\) is located in. Enter the option number that corresponds to your answer.

Option #1: Quadrant I

Option #2: Quadrant II

Option #3: Quadrant III

Option #4: Quadrant IV

(1 point)
\(P'\) is located in Option #
.

Answer 1

To determine where the point \(P(-7, 2)\) will be located after a \(270^\circ\) rotation counterclockwise about the origin, we can use the rotation transformation formula for coordinates.

The rotation transformation for a point \((x, y)\) after a counterclockwise rotation by \(\theta\) degrees is given by:

\[
(x', y') = (x \cos \theta - y \sin \theta, x \sin \theta + y \cos \theta)
\]

In this case, \(\theta = 270^\circ\). We can find the sine and cosine of \(270^\circ\):

\[
\cos(270^\circ) = 0
\]
\[
\sin(270^\circ) = -1
\]

Now, substitute \(x = -7\) and \(y = 2\):

1. For \(x'\):
\[
x' = -7 \cdot 0 - 2 \cdot (-1) = 0 + 2 = 2
\]

2. For \(y'\):
\[
y' = -7 \cdot (-1) + 2 \cdot 0 = 7 + 0 = 7
\]

Thus, the new coordinates after the rotation are:

\[
P'(2, 7)
\]

Now we need to determine the quadrant where the point \(P'(2, 7)\) is located. The coordinates \(x = 2\) and \(y = 7\) are both positive, which means \(P'\) is located in Quadrant I.

Therefore, \(P'\) is located in Option #1: Quadrant I.

The final answer is:

\(P'\) is located in Option #1.