Question

which sequence of transformations will verify that $\triangle xyw$ and $\triangle xyw$ are congruent?
a $\triangle xyw$ is moved onto $\triangle xyw$ after translating $-16$ units vertically, and then reflecting across the $y$-axis.
b $\triangle xyw$ is moved onto $\triangle xyw$ after translating $-16$ units horizontally, and then reflecting across the $y$-axis.
c $\triangle xyw$ is moved onto $\triangle xyw$ after translating $-16$ units vertically, and then reflecting across the $x$-axis.
d $\triangle xyw$ is moved onto $\triangle xyw$ after translating $-16$ units horizontally, and then reflecting across the $x$-axis.

Explanation:

Step1: Analyze horizontal movement

Looking at the \(x -\)coordinates of corresponding vertices (e.g., for \(X\) and \(X'\)). The \(x -\)coordinate of \(X\) is \(10\), and the \(x -\)coordinate of \(X'\) is \(- 6\). The change in the \(x -\)coordinate is \(-6 - 10=-16\). So, there is a horizontal translation of \(- 16\) units (a translation of \(16\) units to the left).

Step2: Analyze reflection

After the horizontal translation, we need to check the reflection. If we consider the \(y -\)coordinates of corresponding vertices (e.g., for \(Y\) after horizontal translation and \(Y'\)). The \(y -\)coordinate of \(Y\) is \(7\), and after horizontal translation (if we assume \(Y\) moves from \(x = 3\) to \(x=3 - 16=-13\)), and then reflecting across the \(y -\)axis. The rule for reflection across the \(y -\)axis is \((x,y)\to(-x,y)\). If we first translate \((x,y)\to(x - 16,y)\) and then reflect \((x - 16,y)\to-(x - 16),y=(16 - x,y)\).

Answer:

B. \(\triangle XYW\) is moved onto \(\triangle X'Y'W'\) after translating \(-16\) units horizontally, and then reflecting across the \(y -\)axis.