Question

triangle abc and triangle xyz are congruent. which sequence of transformations will move triangle abc onto triangle xyz? a reflection across the y-axis, and then a translation 2 units up. a reflection across the x-axis, and then a translation 2 units down. a reflection across the y-axis, and then a translation 2 units right. a reflection across the x-axis, and then a translation 2 units left.

Explanation:

Step1: Analyze reflection across y - axis

First, consider reflecting triangle \(ABC\) across the \(y\) - axis. The rule for reflection across the \(y\) - axis is \((x,y)\to(-x,y)\). For example, if we take a point \(A(- 5,6)\) (assuming approximate coordinates from the graph), after reflection across \(y\) - axis, it becomes \((5,6)\).

Step2: Analyze translation 2 units up

After reflection, we need to translate the triangle 2 units up. The rule for translation 2 units up is \((x,y)\to(x,y + 2)\). If we take the reflected point \((5,6)\), after translating 2 units up, it becomes \((5,8)\), which matches the position of point \(X\) (from the graph of triangle \(XYZ\)). Let's check the other options:

• Option 2: Reflection across \(x\) - axis changes the sign of \(y\) - coordinate, which does not align with the orientation of triangle \(XYZ\) relative to \(ABC\).
• Option 3: Translation 2 units right after reflection across \(y\) - axis would not place the triangle correctly.
• Option 4: Reflection across \(x\) - axis and translation 2 units left also does not match the required transformation.

Answer:

A reflection across the \(y\) - axis, and then a translation 2 units up.