Find a translation that has the same effect as the composition of translations below. \( T_{\langle 1,1\rangle}(x, y) \) followed by \( T_{\langle-2,\rangle\rangle}(x, y) \) Choose the correct answer below. A. \( (x, y) \rightarrow(x+3, y+8) \) B. \( (x, y) \rightarrow(x-1, y-6) \) C. \( (x, y) \rightarrow(x-1, y+8) \) D. \( (x, y) \rightarrow(x+3, y-6) \)

To find the translation that has the same effect as the composition of the translations \( T_{\langle 1,1\rangle}(x, y) \) followed by \( T_{\langle-2,0\rangle}(x, y) \), we will break down the translations step by step. 1. **First Translation**: \( T_{\langle 1,1\rangle}(x, y) \) - This translation moves the point \( (x, y) \) by adding \( 1 \) to the \( x \)-coordinate and \( 1 \) to the \( y \)-coordinate. - The result of this translation is: \[ (x, y) \rightarrow (x + 1, y + 1) \] 2. **Second Translation**: \( T_{\langle-2,0\rangle}(x, y) \) - This translation moves the point \( (x, y) \) by adding \( -2 \) to the \( x \)-coordinate and \( 0 \) to the \( y \)-coordinate. - The result of this translation is: \[ (x, y) \rightarrow (x - 2, y) \] 3. **Composition of Translations**: - Now we apply the second translation to the result of the first translation: \[ (x + 1, y + 1) \rightarrow (x + 1 - 2, y + 1) = (x - 1, y + 1) \] Thus, the overall effect of the composition of the two translations is: \[ (x, y) \rightarrow (x - 1, y + 1) \] Now, we need to find the option that matches this result. Looking at the options: - A. \( (x, y) \rightarrow (x + 3, y + 8) \) - B. \( (x, y) \rightarrow (x - 1, y - 6) \) - C. \( (x, y) \rightarrow (x - 1, y + 8) \) - D. \( (x, y) \rightarrow (x + 3, y - 6) \) None of the options match \( (x - 1, y + 1) \). It seems there might be a mistake in the options provided. However, based on the calculations, the correct translation that has the same effect as the composition of the translations is: \[ (x, y) \rightarrow (x - 1, y + 1) \] If you have any further questions or need clarification, feel free to ask!