Question

which of the following sequences of transformations maps △xyz onto △abc?
sequence a
a translation 2 units to the right and 10 units up, then a reflection over the line $overleftrightarrow{ac}$.
sequence b
a translation 6 units to the right and 6 units up, then a 180° rotation about point b.

Explanation:

Step1: Analyze Sequence A

First, consider the translation in Sequence A. A translation 2 units to the right and 10 units up moves the points of $\triangle XYZ$. Then, a reflection over the line $\overleftrightarrow{AC}$ flips the translated triangle.

Step2: Analyze Sequence B

For Sequence B, a translation 6 units to the right and 6 units up repositions $\triangle XYZ$. Then, a $180^{\circ}$ rotation about point $B$ changes the orientation of the translated - triangle.
Let's assume the coordinates of the vertices of $\triangle XYZ$ are $(x_1,y_1),(x_2,y_2),(x_3,y_3)$. After the translation in Sequence A: The new coordinates become $(x_1 + 2,y_1+10),(x_2 + 2,y_2 + 10),(x_3 + 2,y_3+10)$. Then, after reflection over the line $\overleftrightarrow{AC}$, we use the reflection formula. For a point $(x,y)$ reflected over a line $y=mx + c$, the new - point $(x',y')$ can be found using the appropriate geometric transformation rules.
For Sequence B, after the translation, the coordinates of the vertices of $\triangle XYZ$ become $(x_1 + 6,y_1+6),(x_2 + 6,y_2 + 6),(x_3 + 6,y_3+6)$. Then, for a $180^{\circ}$ rotation about point $B(x_b,y_b)$ of a point $(x,y)$ in the plane, the formula for a $180^{\circ}$ rotation about a point $(a,b)$ is $(2a - x,2b - y)$. Here $a=x_b$ and $b = y_b$.
By visual inspection of the graph and applying the transformation rules:

1. For Sequence A:

• The translation moves $\triangle XYZ$ close to the position of $\triangle ABC$. The reflection over the line $\overleftrightarrow{AC}$ then maps $\triangle XYZ$ onto $\triangle ABC$.

1. For Sequence B:

• The translation moves $\triangle XYZ$ in a different direction. The $180^{\circ}$ rotation about point $B$ does not map $\triangle XYZ$ onto $\triangle ABC$.

Answer:

Sequence A