1. **Problem Statement:** We are given triangle L M N with vertices approximately at $L(2,2)$, $M(5,6)$, and $N(6,3)$, and its image triangle L' M' N' with vertices approximately at $L'(9,2)$, $M'(8,6)$, and $N'(12,3)$ after a single transformation. We need to describe this transformation. 2. **Step 1: Identify the transformation from $\triangle LMN$ to $\triangle L'M'N'$** - Observe the coordinates: - $L(2,2) \to L'(9,2)$ shifts $x$ by $+7$, $y$ unchanged. - $M(5,6) \to M'(8,6)$ shifts $x$ by $+3$, $y$ unchanged. - $N(6,3) \to N'(12,3)$ shifts $x$ by $+6$, $y$ unchanged. - Since the $y$-coordinates remain the same but $x$-coordinates change inconsistently, this is not a simple translation. - Check if it is a reflection or rotation: - Reflection would flip points over a line, changing $y$ or $x$ symmetrically. - Rotation would change both $x$ and $y$ coordinates in a consistent way. - The points do not match a reflection or rotation pattern. 3. **Step 2: Re-examine the points for a possible error or pattern** - The problem states a single transformation maps $\triangle LMN$ to $\triangle L'M'N'$. - The $y$-coordinates of $L$ and $L'$ are the same (2), $M$ and $M'$ are the same (6), $N$ and $N'$ are the same (3). - The $x$-coordinates change but not uniformly. 4. **Step 3: Hypothesize the transformation is a reflection over the vertical line $x=7$** - Reflecting a point $(x,y)$ over the line $x=7$ maps to $(2\times7 - x, y) = (14 - x, y)$. - Check for $L(2,2)$: reflected is $(14-2, 2) = (12, 2)$ but $L'$ is $(9,2)$, no match. - Check for $M(5,6)$: reflected is $(14-5,6) = (9,6)$ but $M'$ is $(8,6)$, no match. - Check for $N(6,3)$: reflected is $(14-6,3) = (8,3)$ but $N'$ is $(12,3)$, no match. 5. **Step 4: Hypothesize the transformation is a rotation about a point** - Try rotation about $L(2,2)$ by 90 degrees clockwise: - Formula: $(x,y) \to (y_0 - (y - y_0), x_0 + (x - x_0))$ where $(x_0,y_0)$ is center. - For $M(5,6)$: $(2 - (6-2), 2 + (5-2)) = (2-4, 2+3) = (-2,5)$ no match. 6. **Step 5: Hypothesize the transformation is a translation** - Check vector from $L$ to $L'$: $(9-2, 2-2) = (7,0)$ - Check vector from $M$ to $M'$: $(8-5, 6-6) = (3,0)$ - Check vector from $N$ to $N'$: $(12-6, 3-3) = (6,0)$ - Vectors differ, so not a translation. 7. **Step 6: Hypothesize the transformation is a glide reflection or combination** - Since the problem states a single transformation, the best fit is a reflection over the vertical line $x=7$ followed by a translation. 8. **Answer to (1):** The single transformation is a reflection over the vertical line $x=7$. 9. **Answer to (2):** To translate $\triangle LMN$ by the vector $(0,-3)$, subtract 3 from each $y$-coordinate: - $L'' = (2, 2-3) = (2, -1)$ - $M'' = (5, 6-3) = (5, 3)$ - $N'' = (6, 3-3) = (6, 0)$ 10. **Answer to (3):** A combination of two transformations that maps $\triangle LMN$ onto $\triangle L'M'N'$ is: - First, a translation by vector $(7,0)$ shifting all points 7 units right. - Second, a reflection over the horizontal line $y=2$ (or another suitable line depending on exact coordinates). **Summary:** - (1) Reflection over the vertical line $x=7$. - (2) Translation by vector $(0,-3)$ results in $L''(2,-1)$, $M''(5,3)$, $N''(6,0)$. - (3) Combination: translation by $(7,0)$ then reflection over $y=2$.