Question

a composition of transformations maps \\(\delta xyz\\) to \\(\delta xyz\\).

the first transformation for this composition is ______, and the second transformation is a \\(90^\circ\\) rotation about point \\(x\\).

• a \\(180^\circ\\) rotation about point \\(x\\)
• a \\(270^\circ\\) rotation about point \\(x\\)
• a translation to the right
• a reflection across line \\(m\\)

Explanation:

Analyze the final state and the second transformation

Using the Composition of Transformations and Geometric Rotations knowledge points

• The final image is \(\Delta X''Y''Z''\).
• The second transformation is a \(90^\circ\) rotation about point \(X'\) to get to \(\Delta X''Y''Z''\).
• Since \(X'\) is the center of rotation, its position does not change during this rotation, meaning \(X' = X''\).
• Looking at the diagram, the point labeled \(X'\) is at the same position as \(X''\).
• Reversing a \(90^\circ\) counterclockwise rotation about \(X'\) by \(90^\circ\) clockwise gives the intermediate state \(\Delta X'Y'Z'\).

Determine the first transformation

Using the Geometric Reflections knowledge point

• We compare the original \(\Delta XYZ\) on the left of line \(m\) with the intermediate \(\Delta X'Y'Z'\).
• Point \(X\) is reflected across line \(m\) to point \(X'\).
• Point \(Y\) is reflected across line \(m\) to point \(Y'\).
• Point \(Z\) is reflected across line \(m\) to point \(Z'\).
• This mapping perfectly matches a reflection across line \(m\).

Answer:

• a 180° rotation about point X
• a 270° rotation about point X
• a translation to the right
• a reflection across line m (Correct answer)