Question

which sequence of transformations will verify that $\triangle efg$ and $\triangle efg$ are congruent?
a $\triangle efg$ is moved onto $\triangle efg$ after rotating $90^\circ$ counterclockwise around the origin and reflecting across the line $x = 3$

b $\triangle efg$ is moved onto $\triangle efg$ after rotating $90^\circ$ counterclockwise around the origin and reflecting across the line $y = 3$

c $\triangle efg$ is moved onto $\triangle efg$ after rotating $180^\circ$ counterclockwise around the origin and reflecting across the line $x = 3$

d $\triangle efg$ is moved onto $\triangle efg$ after rotating $180^\circ$ counterclockwise around the origin and reflecting across the line $y = 3$

Explanation:

Step1: Analyze rotation

A \(180^{\circ}\) counter - clockwise rotation around the origin changes the sign of both \(x\) and \(y\) coordinates of a point \((x,y)\) to \((-x,-y)\). A \(90^{\circ}\) counter - clockwise rotation around the origin changes a point \((x,y)\) to \((-y,x)\). By observing the orientation of \(\triangle EFG\) and \(\triangle E'F'G'\), a \(180^{\circ}\) rotation is more likely as it flips the figure in a way that is more consistent with the overall position change compared to a \(90^{\circ}\) rotation.

Step2: Analyze reflection

After a \(180^{\circ}\) rotation, to map the rotated figure onto \(\triangle E'F'G'\), we consider the reflection. If we look at the vertical position of the figures, reflecting across the line \(y = 3\) (a horizontal line) will adjust the vertical position of the rotated figure to match \(\triangle E'F'G'\). Reflecting across \(x = 3\) (a vertical line) would adjust the horizontal position, which is not what is needed based on the figure's layout.

Answer:

D. \(\triangle EFG\) is moved onto \(\triangle E'F'G'\) after rotating \(180^{\circ}\) counterclockwise around the origin and reflecting across the line \(y = 3\)