The problem describes a transformation of triangle $ABC$ through two steps: a translation followed by a rotation. Let's break down the steps systematically.

Given:

• $A(-1, 3), B(5, -3), C(2, 4)$
• Translation by $T\left(-3, -2\right)$
• Rotation of 90° clockwise about the origin.

Step 1: Translation

To apply the translation $T(-3, -2)$, we add $(-3, -2)$ to each point.

For point $A(-1, 3)$: $A' = (-1 + (-3), 3 + (-2)) = (-4, 1)$

For point $B(5, -3)$: $B' = (5 + (-3), -3 + (-2)) = (2, -5)$

For point $C(2, 4)$: $C' = (2 + (-3), 4 + (-2)) = (-1, 2)$

Step 2: Rotation by 90° Clockwise

To rotate a point $(x, y)$ by 90° clockwise around the origin, the formula is: $(x', y') = (y, -x)$

Applying this to the translated points:

For $A'(-4, 1)$: $A'' = (1, 4)$

For $B'(2, -5)$: $B'' = (-5, -2)$

For $C'(-1, 2)$: $C'' = (2, 1)$

Final coordinates:

• $A''(1, 4)$
• $B''(-5, -2)$
• $C''(2, 1)$

Thus, the correct answer is D: $A''(1,4), B''(-5,-2), C''(2,1)$.

Would you like further details on any of the steps? Here are 5 related questions for deeper understanding:

1. How would the transformation change if the rotation was counterclockwise?
2. What is the general formula for a 90° clockwise rotation around any point other than the origin?
3. Can you calculate the distance between the original and translated points?
4. How does a reflection differ from a rotation in terms of transformation?
5. What is the composition of multiple transformations in geometry?

Tip: When rotating points, always visualize or draw the axes to avoid mistakes!