To solve these questions, let’s address the problem step by step:

Question 1: Determine the Center of Dilation

From the image provided, the dilation transforms triangle $\triangle ABC$ to $\triangle A'B'C'$.

The center of dilation is the fixed point from which the vertices of $\triangle ABC$ are scaled outward (or inward) to their corresponding points in $\triangle A'B'C'$. To find this:

1. Draw lines connecting each pair of corresponding vertices:

   • Line through $A$ and $A'$
   • Line through $B$ and $B'$
   • Line through $C$ and $C'$

2. The intersection point of these lines is the center of dilation.

From the given diagram, these lines intersect at $(0, 1)$.

Answer: The center of dilation is A. (0, 1).

Question 2: Scale Factor of the Dilation

The scale factor $k$ determines how much the triangle was scaled. It is calculated as:

$k = \frac{\text{Distance from center of dilation to a point on } \triangle A'B'C'}{\text{Distance from center of dilation to the corresponding point on } \triangle ABC}$

1. Choose a pair of corresponding points, such as $A$ and $A'$.

   • Distance from center $(0, 1)$ to $A(1, 1)$: $d = \sqrt{(1 - 0)^2 + (1 - 1)^2} = 1$
   • Distance from center $(0, 1)$ to $A'(2, 3)$: $d' = \sqrt{(2 - 0)^2 + (3 - 1)^2} = \sqrt{4 + 4} = 2\sqrt{2}$

2. The scale factor $k = \frac{d'}{d} = \frac{2\sqrt{2}}{1} = 2$.

Answer: The scale factor is C. 2.

If you'd like detailed clarification or have any additional questions, let me know!

5 Related Questions:

1. What are other methods to find the center of dilation geometrically?
2. How do negative scale factors affect dilations on a coordinate plane?
3. What happens when the center of dilation is a vertex of the triangle?
4. How does dilation affect the angles of a figure?
5. Can you determine the scale factor if the original and dilated shapes are given without a graph?

Tip:

Always verify dilation transformations by comparing the ratio of distances from the center to corresponding points; this ensures consistency across all vertices!