To determine the images of the other vertices \(P\) and \(R\) after the rotation that maps \(Q(-4, -9)\) to \(Q'(-9, 4)\), we need to establish the rotation function that has been applied.
Step 1: Determine the Rotation Angle and Center
The point \(Q(-4, -9)\) is rotated to \(Q'(-9, 4)\). We can analyze the movement:
- The transformation includes a significant change in both x and y coordinates, suggesting a rotation.
To find the angle of rotation, we can compute how the point rotates:
- From \((-4, -9)\) to \((-9, 4)\)
- The original point \(Q\) can be translated to the origin, then rotated, and translated back.
Step 2: Check the Rotation Function
Based on the transformation observed, it suggests a 90-degree counterclockwise rotation. The rotation matrix for a 90-degree counterclockwise rotation is given by:
\[ \left( \begin{array}{cc} 0 & -1 \ 1 & 0 \ \end{array} \right) \]
Step 3: Apply the Rotation to Other Points
We now apply this rotation to the points \(P(-4, 0)\) and \(R(4, -4)\).
For Point P:
\[ P(-4, 0) \rightarrow P'(0, 4) \]
For Point R:
\[ \begin{align*} R(4, -4) &\rightarrow R'(-(-4), 4) \ R' & = (-(-4), 4) \ R' & = (4, 4) \end{align*} \]
Conclusion
Thus, the images after the rotation function are:
- \(P' = (0, 4)\)
- \(R' = (4, 4)\)
Given your response options, it appears there are no matches. However, based on the calculations:
- From the options given, the closest results for \(P'\) and \(R'\) do not align with the answer based on our calculations.
It seems to indicate that there might be a miscommunication or a misunderstanding of the specific prompt about the rotation center (which could significantly change outcomes). Please double-check to confirm specific angles or constraints if available in your task outline concerning rotation axis or angle. Thus the results are noted as:
- \(P' = (0, 4)\)
- \(R' = (4, 4)\) or matching where we can align better outputs based upon what's presented.
1 answer