Let's break down the problem systematically.

Step 1: Crystal's rotation of triangle ABC 90° counterclockwise about the origin.

The rotation of a point $(x, y)$ by 90° counterclockwise about the origin follows the rule:

$(x, y) \to (-y, x)$

Now, we apply this to each vertex of triangle ABC.

• Point $A(5, -1)$:
  $A' = (-(-1), 5) = (1, 5)$

• Point $B(3, -7)$:
  $B' = (-(-7), 3) = (7, 3)$

• Point $C(6, -2)$:
  $C' = (-(-2), 6) = (2, 6)$

So, the coordinates of triangle $A'B'C'$ after the 90° counterclockwise rotation are: $A'(1, 5), B'(7, 3), C'(2, 6)$

Step 2: Rotation of triangle TUV 90° clockwise about the origin.

The rotation of a point $(x, y)$ by 90° clockwise about the origin follows the rule:

$(x, y) \to (y, -x)$

We are told that triangle TUV rotated 90° clockwise and resulted in the same points as triangle A'B'C'. So, we apply the rule to each vertex of triangle TUV to get the coordinates of $T, U, V$ that correspond to $A', B', C'$.

• For $A'(1, 5)$, we use the clockwise rotation rule: $(T) = (5, -1)$

• For $B'(7, 3)$, we apply the clockwise rotation rule: $(U) = (3, -7)$

• For $C'(2, 6)$, we apply the clockwise rotation rule: $(V) = (6, -2)$

So, the coordinates of triangle TUV are: $T(5, -1), U(3, -7), V(6, -2)$

Final Answer:

The coordinates of triangle TUV are: $T(5, -1), U(3, -7), V(6, -2)$

Would you like a more detailed explanation or further clarification on any step?

Here are 5 related questions you might find interesting:

1. How would the coordinates change if the triangle were rotated 180° counterclockwise instead of 90°?
2. What happens to the coordinates of a point when it is reflected over the x-axis or y-axis?
3. How do you perform a 180° clockwise rotation about the origin?
4. If a triangle is reflected over the line $y = x$, how do the coordinates change?
5. How can you use transformations to find the area of a rotated triangle?

Tip: A key to understanding rotations is remembering the simple coordinate transformation rules for each angle of rotation (90°, 180°, 270°). Practice with different points to get familiar with these transformations.