To solve the problem of dilating the vertices of the triangle by a scale factor of $\frac{1}{2}$ centered at the origin, let's follow the steps:

Step 1: Identify the Coordinates of the Original Vertices

The vertices of the triangle are labeled as $A$, $B$, and $C$. From the graph, the coordinates appear to be:

• $A (-6, -6)$
• $B (0, -6)$
• $C (-6, -10)$

Step 2: Apply the Dilation Formula

The dilation formula for a point $(x, y)$ with a scale factor $k$ centered at the origin is:

$(x', y') = (k \cdot x, k \cdot y)$

Here, the scale factor $k = \frac{1}{2}$.

Step 3: Calculate the New Coordinates

• For $A (-6, -6)$:

$A' = \left( \frac{1}{2} \cdot (-6), \frac{1}{2} \cdot (-6) \right) = (-3, -3)$

• For $B (0, -6)$:

$B' = \left( \frac{1}{2} \cdot 0, \frac{1}{2} \cdot (-6) \right) = (0, -3)$

• For $C (-6, -10)$:

$C' = \left( \frac{1}{2} \cdot (-6), \frac{1}{2} \cdot (-10) \right) = (-3, -5)$

Step 4: New Coordinates of the Dilated Vertices

• $A' = (-3, -3)$
• $B' = (0, -3)$
• $C' = (-3, -5)$

These are the coordinates of the vertices after dilation with a scale factor of $\frac{1}{2}$.

Would you like further details or clarifications on this solution?

Related Questions:

1. How do you find the image of a point under dilation with a different scale factor?
2. What is the effect of dilation on the area of a triangle?
3. How would the result change if the dilation was centered at a different point?
4. Can dilation change the orientation of a figure?
5. How would a dilation with a scale factor greater than 1 affect the coordinates?

Tip: When dilating around the origin, multiplying each coordinate by the scale factor is all that’s needed!