10. Quadrilateral \( A B C D \) with vertices \( A(2,5) \). \( B(5,8), C(8,-1) \), and \( D(1,1) \) : (a) Translation: \( (x, y) \rightarrow(x-9, y-4) \) (b) Reflection: in the \( y \)-axis

To perform the transformations on the quadrilateral \( ABCD \) with vertices \( A(2,5) \), \( B(5,8) \), \( C(8,-1) \), and \( D(1,1) \), we will follow the steps for each transformation. ### (a) Translation: \( (x, y) \rightarrow (x-9, y-4) \) For the translation, we will subtract 9 from the x-coordinates and 4 from the y-coordinates of each vertex. 1. **Vertex A**: \[ A(2, 5) \rightarrow A'(2-9, 5-4) = A'(-7, 1) \] 2. **Vertex B**: \[ B(5, 8) \rightarrow B'(5-9, 8-4) = B'(-4, 4) \] 3. **Vertex C**: \[ C(8, -1) \rightarrow C'(8-9, -1-4) = C'(-1, -5) \] 4. **Vertex D**: \[ D(1, 1) \rightarrow D'(1-9, 1-4) = D'(-8, -3) \] After the translation, the new vertices are: - \( A'(-7, 1) \) - \( B'(-4, 4) \) - \( C'(-1, -5) \) - \( D'(-8, -3) \) ### (b) Reflection: in the \( y \)-axis For the reflection in the \( y \)-axis, we will change the sign of the x-coordinates of each vertex while keeping the y-coordinates the same. 1. **Vertex A'**: \[ A'(-7, 1) \rightarrow A''(7, 1) \] 2. **Vertex B'**: \[ B'(-4, 4) \rightarrow B''(4, 4) \] 3. **Vertex C'**: \[ C'(-1, -5) \rightarrow C''(1, -5) \] 4. **Vertex D'**: \[ D'(-8, -3) \rightarrow D''(8, -3) \] After the reflection in the \( y \)-axis, the new vertices are: - \( A''(7, 1) \) - \( B''(4, 4) \) - \( C''(1, -5) \) - \( D''(8, -3) \) ### Summary of Results - After translation: - \( A'(-7, 1) \) - \( B'(-4, 4) \) - \( C'(-1, -5) \) - \( D'(-8, -3) \) - After reflection in the \( y \)-axis: - \( A''(7, 1) \) - \( B''(4, 4) \) - \( C''(1, -5) \) - \( D''(8, -3) \)