Question 11 Triangle \( P Q R \) has coordinates \( P(-1,5), Q(1,2) \), and \( R(-3,-1) \). Determine the coordinates of the vertices of the image after a rotation \( 180^{\circ} \) about the origin. \( \begin{array}{ll}\text { A) } P^{\prime}(5,1), Q^{\prime}(2,-1) \text {, and } R^{\prime}(-1,3) \\ \text { B) } P^{\prime}(1,-5), Q^{\prime}(-1,-2) \text {, and } R^{\prime}(3,1) \\ \text { D) } P^{\prime}(-5,-1), Q^{\prime}(-2,1) \text {, and } R^{\prime}(1,-3)\end{array} \)

To find the coordinates of the vertices of triangle \( PQR \) after a rotation of \( 180^\circ \) about the origin, we can use the rule for rotating a point \( (x, y) \) by \( 180^\circ \) around the origin, which is given by: \[ (x, y) \rightarrow (-x, -y) \] Now, we will apply this transformation to each vertex of triangle \( PQR \): 1. For point \( P(-1, 5) \): \[ P' = (-(-1), -5) = (1, -5) \] 2. For point \( Q(1, 2) \): \[ Q' = (-(1), -2) = (-1, -2) \] 3. For point \( R(-3, -1) \): \[ R' = (-(-3), -(-1)) = (3, 1) \] Thus, the coordinates of the vertices after the rotation are: - \( P'(1, -5) \) - \( Q'(-1, -2) \) - \( R'(3, 1) \) Now, let's compare these results with the provided options: - A) \( P'(5, 1), Q'(2, -1), R'(-1, 3) \) - B) \( P'(1, -5), Q'(-1, -2), R'(3, 1) \) - D) \( P'(-5, -1), Q'(-2, 1), R'(1, -3) \) The correct answer is: **B) \( P'(1, -5), Q'(-1, -2), R'(3, 1) \)**.