\( P \) and \( D \) are points on a polygon. \( P^{\prime} \) and \( D^{\prime} \) are the points under a translation. Find \( D^{\prime} \). \( P(3,-2) \) \( P^{\prime}(-10,11) \) \( D(15,4) \)

To find \( D^{\prime} \) under the given translation, follow these steps: 1. **Determine the Translation Vector:** A translation moves every point by the same vector \((a, b)\). Given: \[ P(3, -2) \rightarrow P^{\prime}(-10, 11) \] We can find \(a\) and \(b\) by comparing the coordinates: \[ 3 + a = -10 \quad \Rightarrow \quad a = -13 \] \[ -2 + b = 11 \quad \Rightarrow \quad b = 13 \] So, the translation vector is \((-13, 13)\). 2. **Apply the Translation to Point \( D \):** Given \( D(15, 4) \), apply the translation: \[ D^{\prime} = (15 + a, 4 + b) = (15 - 13, 4 + 13) = (2, 17) \] **Final Answer:** \[ D^{\prime} = (2,\ 17) \]