1. **State the problem:** We need to find the images of the given objects under specified transformations using the diagram and fill in the table. 2. **Given transformations and objects:** - Object A: Rotation by 90° anticlockwise about (0,0) → Image is S (given). - Object P: Reflection in the line $y = -x$ → Find image. - Object P: Translation by $(-12, 2)$ → Image is S (given). - Object T: Rotation by 90° clockwise about (0,0) → Image is P (given). 3. **Recall transformation rules:** - Rotation by 90° anticlockwise about origin: $(x,y) \to (-y,x)$. - Rotation by 90° clockwise about origin: $(x,y) \to (y,-x)$. - Reflection in line $y = -x$: $(x,y) \to (-y,-x)$. - Translation by vector $(a,b)$: $(x,y) \to (x+a,y+b)$. 4. **Find image of P under reflection in $y = -x$:** Given P is around $(-3,-3)$. Apply reflection: $$ (x,y) \to (-y,-x) $$ $$ (-3,-3) \to (-(-3), -(-3)) = (3,3) $$ This matches the location of A. 5. **Summary of images:** - $A \xrightarrow{90^\circ \text{ anticlockwise}} S$ - $P \xrightarrow{\text{reflection in } y=-x} A$ - $P \xrightarrow{\text{translation } (-12,2)} S$ - $T \xrightarrow{90^\circ \text{ clockwise}} P$ 6. **Final table:** | Object | Transformation | Image | |--------|----------------|-------| | A | Rotation by 90° anticlockwise about (0,0) | S | | P | Reflection in the line $y = -x$ | A | | P | Translation by $(-12, 2)$ | S | | T | Rotation by 90° clockwise about (0,0) | P |