Determine a series of transformations that would map Figure H onto Figure I.
Understand the Problem
The problem requires determining a sequence of geometric transformations (such as translation, rotation, reflection, or dilation) or a combination of these that would transform Figure H into Figure I. We need to analyze the positions, orientations, and sizes of the two figures to identify the appropriate transformations.
Answer
A $90^\circ$ clockwise rotation followed by a translation.
Answer for screen readers
A $90^\circ$ clockwise rotation followed by a translation.
Steps to Solve
- Identify the necessary transformations
By observing Figure H and Figure I, we can see that Figure H needs to be rotated and translated to coincide with Figure I. The orientation of Figure H is different from Figure I, suggesting a rotation. Additionally, the location of Figure H is different from Figure I, suggesting a translation.
- Determine the rotation
Visually, Figure H appears to be rotated clockwise by $90^\circ$ or counter-clockwise by $270^\circ$ to align with the orientation of Figure I.
- Determine the translation
After the rotation, we need to translate the figure to its final position. Let's consider the point $(-6, 4)$ in Figure H and after rotation becomes $(4, 6)$. We then need to translate it to $(3, 4)$ in Figure I. This can be achieved by a translation along the vector $\begin{pmatrix} 3-4 \ 4-6 \end{pmatrix} = \begin{pmatrix} -1 \ -2 \end{pmatrix}$.
- Finalize the transformation sequence
Therefore, the sequence of transformations is a $90^\circ$ clockwise rotation followed by a translation.
A $90^\circ$ clockwise rotation followed by a translation.
More Information
Geometric transformations involve changing the position, size, or orientation of a shape. The basic transformations are translation, rotation, reflection, and dilation (scaling). Combinations of these transformations can be used to map one figure onto another.
Tips
- Incorrectly identifying the type of transformation needed (e.g., confusing rotation with reflection).
- Applying transformations in the wrong order, as the order can affect the final result.
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