1. **Stating the problem:** We have triangle vertices at points $A(2,2)$, $B(8,2)$, and $C(5,5)$. We want to find the new coordinates after rotating the triangle $90^\circ$ anticlockwise about the origin. 2. **Formula for rotation:** To rotate a point $(x,y)$ by $90^\circ$ anticlockwise about the origin, use the formula: $$ (x', y') = (-y, x) $$ This means the new $x$ coordinate is the negative of the old $y$, and the new $y$ coordinate is the old $x$. 3. **Applying the rotation to each vertex:** - For $A(2,2)$: $$ A' = (-2, 2) $$ - For $B(8,2)$: $$ B' = (-2, 8) $$ - For $C(5,5)$: $$ C' = (-5, 5) $$ 4. **Explanation:** Rotating $90^\circ$ anticlockwise swaps the coordinates and changes the sign of the original $y$ coordinate to get the new $x$. This preserves distances and angles, so the shape remains congruent. 5. **Final answer:** The new coordinates after rotation are: $$ A'(-2, 2), B'(-2, 8), C'(-5, 5) $$ This matches what you would get by drawing the triangle on graph paper and rotating it using a ruler and protractor.