Let's solve the problem step by step.

Step 1: Rotation by 90° Counterclockwise

To rotate a point $(x, y)$ by 90° counterclockwise around the origin, the new coordinates $(x', y')$ are given by: $(x', y') = (-y, x)$ We need to apply this transformation to the vertices of triangle $\Delta ABC$.

• Point A is at $(-1, 2)$. After rotation: $A' = (-2, -1)$
• Point B is at $(1, -3)$. After rotation: $B' = (3, 1)$
• Point C is at $(4, 2)$. After rotation: $C' = (-2, 4)$

Step 2: Reflection Over the Line $y = -x$

The reflection of a point $(x, y)$ over the line $y = -x$ is given by swapping and negating the coordinates: $(x', y') = (-y, -x)$ Now, apply this transformation to the new vertices $A', B', C'$.

• Point A' is at $(-2, -1)$. After reflection: $A'' = (1, 2)$
• Point B' is at $(3, 1)$. After reflection: $B'' = (-1, -3)$
• Point C' is at $(-2, 4)$. After reflection: $C'' = (-4, 2)$

Final Coordinates of the Transformed Triangle

The new vertices of the transformed triangle are:

• $A'' = (1, 2)$
• $B'' = (-1, -3)$
• $C'' = (-4, 2)$

By checking these points on the coordinate plane, the transformed triangle can be graphed.

Would you like to see further details or explanations?

Here are five questions to expand your understanding:

1. How would the transformation change if the triangle was rotated by 180° instead of 90°?
2. What happens when you reflect a shape over the line $y = x$ instead of $y = -x$?
3. How would the transformation rules change if you reflected over the x-axis?
4. What would happen if the rotation was clockwise instead of counterclockwise?
5. Can you describe the combined transformation as a single matrix operation?

Tip: Remember that combining transformations (like rotation and reflection) can often be simplified by using matrices, which makes it easier to apply multiple transformations at once.