1. **Problem Statement:** We have points $A=(0,1)$, $B=(0,0)$, and $C=(1,0)$ in $\mathbb{R}^2$. An isometry $f$ maps these points to new locations in three scenarios. We want to determine whether $f$ is a translation, rotation, or glide reflection in each case. 2. **Key facts about isometries:** - A **translation** moves every point by the same vector. - A **rotation** preserves distances and angles but changes orientation around a fixed point. - A **glide reflection** is a reflection followed by a translation along the reflection line, reversing orientation. 3. **Scenario (i):** - $f(A)=(1.4,2)$, $f(B)=(1.4,1)$, $f(C)=(2.4,1)$. - Check if $f$ is a translation by finding vectors $\overrightarrow{AB}$ and $\overrightarrow{f(A)f(B)}$: $$\overrightarrow{AB} = (0-0,0-1) = (0,-1)$$ $$\overrightarrow{f(A)f(B)} = (1.4-1.4,1-2) = (0,-1)$$ - Similarly, check $\overrightarrow{BC}$ and $\overrightarrow{f(B)f(C)}$: $$\overrightarrow{BC} = (1-0,0-0) = (1,0)$$ $$\overrightarrow{f(B)f(C)} = (2.4-1.4,1-1) = (1,0)$$ - Since vectors between points are preserved, $f$ acts like a translation. - To confirm, check if the translation vector is the same for all points: $$\overrightarrow{AA'} = (1.4-0,2-1) = (1.4,1)$$ $$\overrightarrow{BB'} = (1.4-0,1-0) = (1.4,1)$$ $$\overrightarrow{CC'} = (2.4-1,1-0) = (1.4,1)$$ - All translation vectors match, so $f$ is a translation. - It is not a glide reflection because glide reflections reverse orientation, but here orientation is preserved. 4. **Scenario (ii):** - $f(A)=(0.4,1.8)$, $f(B)=(1,1)$, $f(C)=(1.8,1.6)$. - Check if $f$ is a translation by comparing translation vectors: $$\overrightarrow{AA'} = (0.4-0,1.8-1) = (0.4,0.8)$$ $$\overrightarrow{BB'} = (1-0,1-0) = (1,1)$$ - Vectors differ, so $f$ is not a translation. - Check if $f$ is a glide reflection by checking orientation: - Original triangle $ABC$ has vertices in order $A \to B \to C$. - Image points $f(A), f(B), f(C)$ are not in the same orientation order (verified by cross product sign). - Since $f$ does not preserve orientation and is not a translation or glide reflection, it must be a rotation. 5. **Scenario (iii):** - $f(A)=(1.8,1.6)$, $f(B)=(1,1)$, $f(C)=(0.4,1.8)$. - Check orientation: - The order of points is reversed compared to original $ABC$. - Since orientation is reversed and points are not simply translated, $f$ is a glide reflection. **Final answers:** - (i) $f$ is a translation. - (ii) $f$ is a rotation. - (iii) $f$ is a glide reflection.