Square \( A B C D \) is located on a coordinate plane. The coordinates for three of the vertices are listed below. - A \( (2,7) \) - C \( (8,1) \) - D \( (2,1) \) Square \( A B C D \) is dilated by a scale factor of 2 with the center of dilation at the origin, to form square \( A^{\prime} B^{\prime} C^{\prime} D^{\prime} \). What are the coordinates of vertex \( B^{\prime} \) ? Explain how you determined your answer.

To find the coordinates of vertex \( B^{\prime} \), let's first determine the coordinates of vertex \( B \) of square \( ABCD \). Given the coordinates of points A, C, and D, we can find B as follows: 1. Since \( D \) is directly below \( A \) (both have the same x-coordinate), we can find the y-coordinate of \( B \). The y-coordinate of \( B \) will be equal to that of \( C \) because it's diagonally opposite to \( A \) in the square. 2. The sides of the square are equal, so the distance between \( A \) and \( D \) must be the same as the distance between \( A \) and \( B \). The coordinates of \( A \) are \( (2, 7) \) and those of \( D \) are \( (2, 1) \), hence the distance \( AD = 7 - 1 = 6 \). 3. To find point \( B \), we can calculate the length of line segment \( AC \) which connects \( A \) and \( C \). The x-coordinates change from 2 to 8, hence the distance along the x-axis is \( 8 - 2 = 6 \) and the distance along the y-axis is \( 7 - 1 = 6 \). Now, point \( B \) must consequently be at \( (8, 7) \). Next, we apply the dilation with a scale factor of 2. 4. Dilation from the origin means scaling the coordinates of \( B \) by 2. For the coordinates \( B (8, 7) \): - New x-coordinate: \( 8 \times 2 = 16 \) - New y-coordinate: \( 7 \times 2 = 14 \) Therefore, the coordinates of vertex \( B^{\prime} \) after dilation are \( (16, 14) \).