14. If a figure is translated with the rule \( T_{\langle-5,3\rangle} \), which translation moves the image back to the original position? (A) \( T_{\langle 5,-3\rangle} \) (B) \( T_{\langle-5,3\rangle} \) (C) \( T_{\langle 0,5\rangle} \) (D) \( T_{\langle-5,0\rangle} \)

To determine which translation moves the image back to the original position after a translation defined by \( T_{\langle-5,3\rangle} \), we need to understand how translations work. 1. **Understanding the Translation**: The translation \( T_{\langle-5,3\rangle} \) means that every point of the figure is moved 5 units to the left (negative x-direction) and 3 units up (positive y-direction). 2. **Finding the Inverse Translation**: To move the image back to its original position, we need to apply the opposite translation. The opposite of moving left by 5 units is moving right by 5 units, and the opposite of moving up by 3 units is moving down by 3 units. 3. **Calculating the Inverse Translation**: Therefore, the translation that would move the image back to the original position is: \[ T_{\langle 5, -3 \rangle} \] 4. **Identifying the Correct Option**: Now, we can compare this with the provided options: - (A) \( T_{\langle 5,-3\rangle} \) - This is the correct translation. - (B) \( T_{\langle-5,3\rangle} \) - This is the original translation. - (C) \( T_{\langle 0,5\rangle} \) - This does not move the figure back to the original position. - (D) \( T_{\langle-5,0\rangle} \) - This also does not move the figure back to the original position. Thus, the correct answer is: **(A) \( T_{\langle 5,-3\rangle} \)**.