Question: Write the coordinates of the vertices after a dilation with a scale factor of $\frac{1}{2}$, centered at the origin.
A' ($__, __$)
1. **Problem Statement:**
We are given a triangle with vertices $A(-5, -6)$, $B(0, -6)$, and $C(0, -10)$.
We need to find the coordinates of the vertices after a dilation with scale factor $\frac{1}{2}$ centered at the origin.
2. **Formula for Dilation:**
If a point $P(x, y)$ is dilated by a scale factor $k$ centered at the origin, the image point $P'(x', y')$ is given by:
$$
x' = kx, \quad y' = ky
$$
3. **Apply the formula to each vertex:**
- For $A(-5, -6)$:
$$
A' = \left( \frac{1}{2} \times (-5), \frac{1}{2} \times (-6) \right) = (-2.5, -3)
$$
- For $B(0, -6)$:
$$
B' = \left( \frac{1}{2} \times 0, \frac{1}{2} \times (-6) \right) = (0, -3)
$$
- For $C(0, -10)$:
$$
C' = \left( \frac{1}{2} \times 0, \frac{1}{2} \times (-10) \right) = (0, -5)
$$
4. **Final Answer:**
The coordinates of the vertices after dilation are:
$$
A'(-2.5, -3), \quad B'(0, -3), \quad C'(0, -5)
$$