Question

7 multiple choice 6 points prove that triangle abc is congruent to def using a sequence of rigid transformations. rotation 180 degrees about the origin reflection over the y - axis then translations 1 unit left and 2 units up rotation 180 degrees about the origin and then a reflection translation 4 units to the right and 2 units up followed by a reflection over the x axis

Explanation:

Step1: Analyze rigid - transformations

Rigid transformations include rotations, reflections, and translations. We need to check each option to see if it maps $\triangle ABC$ to $\triangle DEF$.

Step2: Check rotation 180 - degrees about origin

A 180 - degree rotation about the origin changes the sign of both the $x$ and $y$ coordinates of each point. This will not map $\triangle ABC$ to $\triangle DEF$.

Step3: Check reflection over y - axis then translations

A reflection over the $y$-axis changes the sign of the $x$ - coordinate of each point. Then, a translation of 1 unit left and 2 units up will not map $\triangle ABC$ to $\triangle DEF$.

Step4: Check rotation 180 degrees about origin and then reflection

A 180 - degree rotation about the origin followed by a reflection will not map $\triangle ABC$ to $\triangle DEF$.

Step5: Check translation and reflection

If we translate $\triangle ABC$ 4 units to the right and 2 units up, and then reflect over the $x$ - axis, we can map $\triangle ABC$ to $\triangle DEF$.
Let's assume a general point $(x,y)$ in $\triangle ABC$. After translation 4 units right and 2 units up, the point becomes $(x + 4,y+2)$. After reflection over the $x$ - axis, the point becomes $(x + 4,-(y + 2))$. This sequence of transformations maps $\triangle ABC$ to $\triangle DEF$.

Answer:

Translation 4 units to the right and 2 units up followed by a reflection over the x - axis