Question

a composition of transformations maps $\triangle xyz$ to $\triangle xyz$.
the first transformation for this composition is \\_____, and the second transformation is a $90^\circ$ rotation about point $x$.
options:

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

Explanation:

Step1: Analyze the first transformation

To determine the first transformation, we observe the position of \(\triangle XYZ\) and its image before the rotation. A reflection across line \(m\) (the red line) would map \(\triangle XYZ\) to a position that aligns with the pre - rotation image of \(\triangle XY''Z''\) relative to point \(X\). A translation or rotation about \(X\) at this stage does not match the initial mapping as well as a reflection across line \(m\).

Step2: Confirm the second transformation (given as 90° rotation about \(X\), but we focus on the first)

From the options, the first transformation that makes sense for the composition (to get to the stage before the 90° rotation about \(X\)) is a reflection across line \(m\). Wait, no, let's re - evaluate. Wait, the first transformation: looking at the triangle \(\triangle XYZ\) and the intermediate triangle (before the 90° rotation about \(X\)), the first transformation is a reflection across line \(m\)? Wait, no, the options for the first transformation are "a translation to the right", "a reflection across line \(m\)", "a 270° rotation about point \(X\)", "a 180° rotation about point \(X\)".

Wait, let's look at the triangles. The original triangle \(\triangle XYZ\) and the triangle after the first transformation (before the 90° rotation about \(X\)). A reflection across line \(m\) (the horizontal red line) would flip the triangle over line \(m\), which seems to be the case here. Alternatively, let's check the other options. A translation to the right would move the triangle without flipping, but the orientation changes. A 270° or 180° rotation about \(X\) would rotate the triangle around \(X\), but the initial mapping from \(\triangle XYZ\) to the next triangle (before the 90° rotation) is a reflection across line \(m\)? Wait, no, maybe I made a mistake. Wait, the first transformation is the one that maps \(\triangle XYZ\) to the triangle that is then rotated 90° about \(X\) to get \(\triangle XY''Z''\). So the first transformation: among the options, the correct first transformation is a reflection across line \(m\)? Wait, no, the options are:

1. a translation to the right
2. a reflection across line \(m\)
3. a 270° rotation about point \(X\)
4. a 180° rotation about point \(X\)

Wait, let's look at the triangle \(\triangle XYZ\) and the triangle with vertices \(X\), \(Y'\), \(Z'\). The transformation from \(\triangle XYZ\) to \(\triangle XY'Z'\) is a reflection across line \(m\) (the red line). So the first transformation is a reflection across line \(m\). But wait, the problem says "The first transformation for this composition is [ ] and the second transformation is a 90° rotation about point \(X\)".

Wait, maybe I misread. Let's check the options again. The first transformation options:

• a translation to the right: would move the triangle horizontally, but the triangle's orientation and position relative to line \(m\) suggests a reflection.
• a reflection across line \(m\): this flips the triangle over line \(m\), which matches the visual change from \(\triangle XYZ\) to the triangle below it (before the 90° rotation about \(X\)).
• a 270° rotation about point \(X\): rotating 270° about \(X\) would move the triangle in a circular motion, not matching the flip over line \(m\).
• a 180° rotation about point \(X\): rotating 180° about \(X\) would turn the triangle upside - down relative to \(X\), not matching the flip over line \(m\).

Answer:

The first transformation is a reflection across line \(m\) (the option "a reflection across line \(m\)").