This problem involves two transformations of the triangle:

Step 1: Rotation 90° counterclockwise about the origin

To perform this rotation, you apply the transformation rule:

• If a point on the triangle has coordinates $(x, y)$, after a 90° counterclockwise rotation, the new coordinates will be: $(x, y) \to (-y, x)$

For example, if the vertices of the original triangle are:

• $A(x_1, y_1)$,
• $B(x_2, y_2)$,
• $C(x_3, y_3)$,

then after the rotation, their new positions will be:

• $A'(-y_1, x_1)$,
• $B'(-y_2, x_2)$,
• $C'(-y_3, x_3)$.

Step 2: Dilation with a scale factor of $\frac{1}{2}$ centered at the origin

To perform the dilation, you multiply the coordinates of each vertex by the scale factor $\frac{1}{2}$. The rule is:

• If the new coordinates after rotation are $(x', y')$, the dilated coordinates will be: $(x', y') \to \left(\frac{x'}{2}, \frac{y'}{2}\right)$

This reduces the size of the triangle by half while keeping it centered at the origin.

Procedure:

1. Write down the coordinates of the vertices of the given triangle.
2. Apply the rotation rule to each vertex to get the new positions.
3. Use the dilation rule to scale the rotated triangle down by half.

Would you like help identifying the coordinates of the triangle's vertices and working through these steps in detail? Let me know if you’d like a step-by-step solution.

Related Questions:

1. What are the rules for performing rotations of 180° or 270°?
2. How does dilation affect the area of a shape?
3. What is the difference between dilation and translation?
4. How do you determine the scale factor from a dilation?
5. How does the center of dilation affect the transformation?

Tip: Always visualize transformations step-by-step on graph paper to confirm your results!