A composition of transformations maps $\triangle XYZ$ to $\Delta X^{"'} Y^{"'} Z^{"''}$. The first transformation for this composition is
[BLANK], and the second transformation is a $90^{\circ}$ rotation about point $X^{\prime}$.
A. a $180^{\circ}$ rotation about point X
B. a $270^{\circ}$ rotation about point $X$
C. a translation to the right
D. a reflection across line $m$
Answer (1)
Without more information about the final image △ X ′′ Y ′′ Z ′′ , we cannot determine which of the four options is the correct first transformation. Each of the options is a valid first transformation.
Explanation
Analyze the options Let's analyze the given options for the first transformation and how they affect the position of point X to its image X ′ . The second transformation is always a 9 0 ∘ rotation about X ′ .
Option 1: a 18 0 ∘ rotation about point X In this case, X ′ would coincide with X . So, the second transformation is a 9 0 ∘ rotation about point X .
Option 2: a 27 0 ∘ rotation about point X In this case, X ′ would coincide with X . So, the second transformation is a 9 0 ∘ rotation about point X .
Option 3: a translation to the right In this case, X ′ would be located to the right of X . So, the second transformation is a 9 0 ∘ rotation about point X ′ .
Option 4: a reflection across line m In this case, X ′ would be the reflection of X across line m . So, the second transformation is a 9 0 ∘ rotation about point X ′ .
Without additional information about the final image △ X ′′ Y ′′ Z ′′ , it is impossible to determine which of the four options is the correct first transformation. Each of the options is a valid first transformation.
Examples
Understanding transformations is crucial in fields like computer graphics and robotics. For instance, when programming a robot to assemble a product, you need to apply a series of transformations (rotations, translations, reflections) to move parts from one location to another. Each transformation must be precise to ensure the final product is assembled correctly. This problem illustrates the basic concept of combining transformations, which is a fundamental skill in these areas.
