Question

which transformation proves △abc≅△def? o rotation of of △abc 90° clockwise, centered at the origin o translation of △abc 7 units right and 9 units down o reflection of △abc over the y - axis and then over the x - axis o reflection of △abc over the line y = x

Explanation:

Step1: Analyze rotation

A 90 - degree clock - wise rotation centered at the origin changes the coordinates of a point $(x,y)$ to $(y, - x)$. By observing the positions of the vertices of $\triangle ABC$ and $\triangle DEF$, a 90 - degree clock - wise rotation of $\triangle ABC$ centered at the origin will not map $\triangle ABC$ onto $\triangle DEF$.

Step2: Analyze translation

A translation of $\triangle ABC$ 7 units right and 9 units down changes the coordinates of a point $(x,y)$ to $(x + 7,y - 9)$. By observing the graph, this translation will not map $\triangle ABC$ onto $\triangle DEF$.

Step3: Analyze double - reflection

A reflection of $\triangle ABC$ over the y - axis changes the coordinates of a point $(x,y)$ to $(-x,y)$, and then a reflection over the x - axis changes $(-x,y)$ to $(-x,-y)$. This double - reflection will not map $\triangle ABC$ onto $\triangle DEF$.

Step4: Analyze reflection over $y = x$

A reflection of a point $(x,y)$ over the line $y=x$ changes it to $(y,x)$. By observing the vertices of $\triangle ABC$ and $\triangle DEF$, a reflection of $\triangle ABC$ over the line $y = x$ will map $\triangle ABC$ onto $\triangle DEF$.

Answer:

reflection of $\triangle ABC$ over the line $y = x$