Question

given $\triangle def \cong \triangle def$, describe a sequence of transformations that maps $\triangle def$ to $\triangle def$.

choose the correct answer below.

a. translation of 6 units down, translation of 6 units right, reflection across the y - axis
b. reflection across the x - axis, translation of 6 units left, translation of 6 units up
c. reflection across the y - axis, reflection across the x - axis, translation of 6 units left
d. rotation of $90^\circ$ about the origin, translation of 6 units down, translation of 6 units left

Explanation:

Step1: Analyze Reflection Across x - axis

First, consider reflecting \(\triangle DEF\) across the \(x\) - axis. A reflection across the \(x\) - axis changes the sign of the \(y\) - coordinate of each point \((x,y)\) to \((x, - y)\). This will flip the triangle over the \(x\) - axis, aligning its orientation more closely to \(\triangle D'E'F'\).

Step2: Analyze Translation of 6 Units Up

After the reflection across the \(x\) - axis, we need to translate the triangle 6 units up. A translation of 6 units up adds 6 to the \(y\) - coordinate of each point \((x,y)\) (after reflection) to get \((x,y + 6)\). This translation, combined with the reflection, will map \(\triangle DEF\) to \(\triangle D'E'F'\). Let's check the other options:

• Option A: A translation of 6 units down and then 6 units right will not map \(\triangle DEF\) to \(\triangle D'E'F'\) as the orientation and position changes are incorrect.
• Option C: Reflecting across the \(y\) - axis and then \(x\) - axis and translating left will not give the correct mapping.
• Option D: A rotation of \(90^{\circ}\) about the origin and then translations will not match the transformation from \(\triangle DEF\) to \(\triangle D'E'F'\).

Answer:

B. Reflection across the x - axis, translation of 6 units up