Question
\( P \) and \( D \) are points on a polygon. \( P^{\prime} \) and \( D^{\prime} \) are the points under a translation. Find \( D^{\prime} \). \( P(3,-2) \) \( P^{\prime}(-10,11) \) \( D(15,4) \)
Ask by Hobbs Joseph. in the United States
Feb 03,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
\( D^{\prime} = (2, 17) \)
Solution
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:**
\[
D^{\prime} = (2,\ 17)
\]
Reviewed and approved by the UpStudy tutoring team
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Simplify this solution Beyond the Answer
To find \( D^{\prime} \) under the same translation that moves \( P \) to \( P^{\prime} \), we first determine the translation vector. We can find this by subtracting the coordinates of \( P \) from \( P^{\prime} \): \[ \text{Translation vector} = P^{\prime} - P = (-10, 11) - (3, -2) = (-10 - 3, 11 + 2) = (-13, 13). \] Now, we apply this translation vector to point \( D \): \[ D^{\prime} = D + \text{Translation vector} = (15, 4) + (-13, 13) = (15 - 13, 4 + 13) = (2, 17). \] Thus, the coordinates of \( D^{\prime} \) are \( \boxed{(2, 17)} \).
