A picture is shown of a square with its corners labeled, A, B, C, and D. A is shown to be located at coordinates, (-5, 4) while B is shown to be located at (-1, 4). The locations of C and D are not specified.

Question

After applying a translation vector to rectangle ABDC , the coordinates of A′ are (−6,8) . What are the coordinates of B′ ?

Answer 1

To find the coordinates of \( B' \) after the same translation that moved \( A \) to \( A' \), we first need to determine the translation vector.

The original coordinates of point \( A \) are: \[ A(-5, 4) \]

The new coordinates of point \( A' \) after the translation are: \[ A'(-6, 8) \]

To find the translation vector \( T \), we can calculate the change in coordinates for point \( A \) to get from \( A \) to \( A' \): \[ T_x = x' - x = -6 - (-5) = -6 + 5 = -1 \] \[ T_y = y' - y = 8 - 4 = 4 \]

Thus, the translation vector is: \[ T = (-1, 4) \]

Now, we apply this translation vector \( T \) to point \( B \). The original coordinates of point \( B \) are: \[ B(-1, 4) \]

Now let's calculate the new coordinates \( B' \): \[ B'_x = B_x + T_x = -1 + (-1) = -1 - 1 = -2 \] \[ B'_y = B_y + T_y = 4 + 4 = 8 \]

Therefore, the coordinates of point \( B' \) after the translation are: \[ B'(-2, 8) \]