Let's solve the given problem.

Problem Details:

• Original vertex $A$ is located at $(6, 4)$.
• Triangle $ABC$ is dilated with a scale factor of $0.5$, with the center of dilation at the origin $(0, 0)$.
• The question asks for the new coordinates of $A'$ after dilation.

Solution:

1. The formula for dilation when the center is at the origin is: $A'(x', y') = (k \cdot x, k \cdot y)$ where:

   • $(x, y)$ are the original coordinates,
   • $k$ is the scale factor.

2. Substitute the given values:

   • $k = 0.5$,
   • $(x, y) = (6, 4)$.

   Perform the calculations: $x' = 0.5 \cdot 6 = 3, \quad y' = 0.5 \cdot 4 = 2.$

3. The new coordinates of $A'$ are $(3, 2)$.

Final Answer:

The coordinates of vertex $A'$ are (3, 2), which corresponds to option A.

Would you like a detailed explanation of the dilation concept? Let me know! Here are some related questions for practice:

1. How do you calculate the coordinates of a point after a dilation with a scale factor greater than 1?
2. What happens to the shape of a figure when the scale factor is less than 1?
3. How do you determine the center of dilation if it is not the origin?
4. How is the area of a figure affected by a dilation?
5. Can a dilation ever change the orientation of a shape?

Tip:

Always multiply each coordinate by the scale factor to ensure the dilation transformation is applied correctly.