Question

analyzing a sequence of transformations
kyta applied ( r_m circ t_{0,3} ) to triangle abc. what mistakes did she make? choose two correct answers.
she applied a rotation instead of a reflection.
she applied the rotation to the triangle first.
she applied the reflection to the triangle first.
she translated the triangle down instead of just applying a reflection.
she changed the size of the triangle instead of just applying a translation.

Explanation:

1. For the composition \( r_{m}\circ T_{0,3} \), the notation means first apply the translation \( T_{0,3} \) (translate 0 units horizontally and 3 units vertically, or as per context) then apply the reflection \( r_{m} \) over line \( m \).

• "She applied the rotation instead of a reflection": The \( r_{m} \) is a reflection, not rotation, so if she did rotation, that's a mistake. But also, the order: composition \( f\circ g \) means \( g \) first then \( f \). So \( r_{m}\circ T_{0,3} \) is translation first, then reflection.
• "She applied the rotation to the triangle first": Wait, no, \( r_{m}\circ T_{0,3} \) is translation (\( T_{0,3} \)) first, then reflection (\( r_{m} \)). If she did rotation (wrong transformation) or applied reflection first (wrong order), but looking at the options:
• "She applied the rotation instead of a reflection": \( r_{m} \) is reflection, so if she used rotation, that's a mistake.
• "She applied the rotation to the triangle first": Wait, maybe mislabeled, but the key is composition order: \( r_{m}\circ T_{0,3} = r_{m}(T_{0,3}(\triangle ABC)) \), so translation first, then reflection. If she did reflection first (applied \( r_{m} \) first) or used rotation, or other errors.
• Looking at the options, the two correct ones are:
• "She applied a rotation instead of a reflection" (since \( r_{m} \) is reflection, not rotation)
• "She applied the rotation to the triangle first" (actually, the composition is translation first, then reflection; if she did rotation (wrong op) first, or maybe the option is about order: if she applied reflection first (but the option says rotation, maybe typo, but more likely: the composition is \( T \) then \( r \), so if she did \( r \) first (applied reflection first) but the option says "rotation" – maybe the first mistake is using rotation instead of reflection, and the second is applying the (wrong) rotation first (instead of translation first). Also, "She applied the rotation to the triangle first" – since the composition is translation first, then reflection, so if she did rotation (wrong) first, that's a mistake. Also, "She applied a rotation instead of a reflection" (correct, as \( r_m \) is reflection).

Wait, re-examining: The transformation is \( r_{m}\circ T_{0,3} \), which is "reflection over \( m \) after translation \( T_{0,3} \)". So steps: 1. Translate \( \triangle ABC \) by \( T_{0,3} \) to get \( \triangle A'B'C' \), 2. Reflect \( \triangle A'B'C' \) over line \( m \) to get \( \triangle A''B''C'' \).

Now, check the options:

• "She applied a rotation instead of a reflection": If she used rotation (like 90 degrees, etc.) instead of reflection \( r_m \), that's a mistake.
• "She applied the rotation to the triangle first": The composition is translation first, then reflection. If she did rotation (wrong op) first, that's a mistake (order wrong, and transformation wrong).
• "She applied the reflection to the triangle first": No, the composition is translation first.
• "She translated the triangle down instead of just applying a reflection": The transformation is a composition, not just reflection.
• "She changed the size of the triangle instead of just applying a translation": The triangles look same size, so no.

So the two mistakes are:

1. She applied a rotation instead of a reflection (since \( r_m \) is reflection, not rotation).
2. She applied the rotation to the triangle first (since the composition is translation first, then reflection; so she did the (wrong) rotation first instead of translation).

Answer:

A. She applied a rotation instead of a reflection
B. She applied the rotation to the triangle first