Triangle \( P Q R \) has coordinates \( P(-1,5), Q(1,2) \), and \( P(-3,-1) \). Determine the coordinates of the vertices of the image after a rotation goof counterclockwise about the origin. \( \begin{array}{ll}\text { A) } P(5,1), Q^{\prime}(2,-1) \text {, and } R^{\prime}(-1,3) \\ \begin{array}{ll}\text { B) } P^{\prime}(-1,-5), Q^{\prime}(1,-2) \text {, and } R^{\prime}(-3,1)\end{array} \\ \begin{array}{ll}\text { C) } P^{\prime}(-5,1), Q^{\prime}(-2,-1) \text {, and } R^{\prime}(1,3)\end{array}\end{array} \)

To find the coordinates of the vertices of triangle \( PQR \) after a counterclockwise rotation about the origin, we can use the rotation transformation formulas. A point \( (x, y) \) rotated counterclockwise by an angle \( \theta \) around the origin transforms to: \[ (x', y') = (x \cos \theta - y \sin \theta, x \sin \theta + y \cos \theta) \] For a 90-degree (or \( \frac{\pi}{2} \) radians) counterclockwise rotation, the cosine and sine values are: \[ \cos 90^\circ = 0, \quad \sin 90^\circ = 1 \] Thus, the transformation simplifies to: \[ (x', y') = (-y, x) \] Now, we will apply this transformation to each vertex of triangle \( PQR \): 1. **Point \( P(-1, 5) \)**: \[ P' = (-5, -1) \] 2. **Point \( Q(1, 2) \)**: \[ Q' = (-2, 1) \] 3. **Point \( R(-3, -1) \)**: \[ R' = (1, -3) \] Now we have the transformed coordinates: - \( P'(-5, -1) \) - \( Q'(-2, 1) \) - \( R'(1, -3) \) Now, let's compare these results with the provided options: - **A)** \( P(5,1), Q^{\prime}(2,-1), R^{\prime}(-1,3) \) - **B)** \( P^{\prime}(-1,-5), Q^{\prime}(1,-2), R^{\prime}(-3,1) \) - **C)** \( P^{\prime}(-5,1), Q^{\prime}(-2,-1), R^{\prime}(1,3) \) None of the options match the calculated coordinates. Let's double-check the calculations: 1. For \( P(-1, 5) \): \[ P' = (-5, -1) \] 2. For \( Q(1, 2) \): \[ Q' = (-2, 1) \] 3. For \( R(-3, -1) \): \[ R' = (1, -3) \] The calculations are correct, but it seems there was a misunderstanding in the options provided. The correct transformed coordinates are: - \( P'(-5, -1) \) - \( Q'(-2, 1) \) - \( R'(1, -3) \) Since none of the options match, it appears there may be an error in the options given.