Question

p(-7, 2) is rotated 270 (counterclockwise) about the origin. name the quadrant P' is located in.

Option #1: Quadrant I

Option #2: Quadrant II

Option #3: Quadrant III

Option #4: Quadrant IV

Answer 1

To determine the new location of the point \( P(-7, 2) \) after a 270-degree counterclockwise rotation about the origin, we can use the rotation transformation for coordinates.

The general formula for a counterclockwise rotation of a point \( (x, y) \) by an angle \( \theta \) is given by:

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

For a 270-degree rotation, we know:

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

Thus, the transformation becomes:

\[
(x', y') = (x \cdot 0 - y \cdot (-1), x \cdot (-1) + y \cdot 0) = (y, -x)
\]

Now we apply this to the point \( P(-7, 2) \):

1. \( x = -7 \)
2. \( y = 2 \)

Calculating the new coordinates:

\[
x' = y = 2
\]
\[
y' = -x = -(-7) = 7
\]

So the new point after rotation is \( P'(2, 7) \).

Now we can determine the quadrant in which \( P'(2, 7) \) lies:

- \( x' = 2 \) (positive)
- \( y' = 7 \) (positive)

Since both coordinates are positive, the point \( P'(2, 7) \) is located in **Quadrant I**.

Thus, the answer is:

**Option #1: Quadrant I**.