Triangle Translation 6601A7

1. **State the problem:** We have a triangle $L\!M\!N$ in the top-left quadrant and its image $L''\!M''\!N''$ in the bottom-right quadrant. We need to find the translation rule that maps the preimage points $(x,y)$ to the image points $(x',y')$. 2. **Recall translation formula:** A translation moves every point by adding a fixed amount to the $x$ and $y$ coordinates: $$ (x,y) \rightarrow (x + a, y + b) $$ where $a$ is the horizontal shift and $b$ is the vertical shift. 3. **Analyze quadrant positions:** - The preimage is in the top-left quadrant, so $x$ is negative or small, $y$ is positive. - The image is in the bottom-right quadrant, so $x'$ is positive, $y'$ is negative. 4. **Determine shifts:** - To move from left to right, $x$ must increase by a positive number $a$. - To move from top to bottom, $y$ must decrease by a positive number $|b|$ (so $b$ is negative). 5. **Check options:** - $(x,y)\rightarrow(x+6,y-5)$: $x$ increases by 6, $y$ decreases by 5. - $(x,y)\rightarrow(x-6,y+5)$: $x$ decreases by 6, $y$ increases by 5. - $(x,y)\rightarrow(x-5,y+6)$: $x$ decreases by 5, $y$ increases by 6. - $(x,y)\rightarrow(x+5,y-6)$: $x$ increases by 5, $y$ decreases by 6. 6. **Choose the correct translation:** Since the image is to the right and down from the preimage, the translation must increase $x$ and decrease $y$. Both $(x+6,y-5)$ and $(x+5,y-6)$ fit this pattern. 7. **Final step:** Without exact coordinates, we compare the magnitude of shifts. The problem states the image is in the bottom-right quadrant, so the translation with larger horizontal shift and smaller vertical shift is more plausible. **Answer:** The translation is $$ (x,y) \rightarrow (x+6, y-5) $$