Question

a composition of transformations maps \\( \triangle xyz \\) to \\( \triangle x^{\prime \prime}y^{\prime \prime}z^{\prime \prime} \\).

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

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

Explanation:

• A translation moves a figure without rotating or resizing it. A reflection flips a figure over a line. A rotation turns a figure around a point.
• Looking at the triangles, first, we observe that the orientation of \(\triangle XYZ\) changes in a non - rotational (in terms of the initial transformation before the \(90^{\circ}\) rotation about \(X'\)) and non - translational (since translation would just shift the figure without flipping) way. A reflection across a line (in this case, line \(m\)) changes the orientation of the figure. After reflection across line \(m\) (mapping \(X\) to \(X'\)), then a \(90^{\circ}\) rotation about \(X'\) can map the reflected triangle to \(\triangle X''Y''Z''\).
• A \(180^{\circ}\) or \(270^{\circ}\) rotation about \(X\) would not give the intermediate figure (before the \(90^{\circ}\) rotation about \(X'\)) as the position of \(X\) relative to the other vertices changes in a non - rotational (for the first transformation) way. A translation to the right would not change the orientation of the triangle in the way shown in the diagram.

Answer:

a reflection across line \(m\)