To find the coordinates of the center of enlargement, we need to understand the geometric transformation, specifically the concept of enlargement with a given scale factor. Let's denote the center of enlargement as [tex]\( C(x, y) \)[/tex].
Given:
- Original point [tex]\( A(2, 4) \)[/tex]
- Image point [tex]\( A'(-1, -2) \)[/tex]
- Scale factor [tex]\( k = -2 \)[/tex]
The formula relating the original point [tex]\( A \)[/tex], the center of enlargement [tex]\( C \)[/tex], the image point [tex]\( A' \)[/tex], and the scale factor [tex]\( k \)[/tex] is:
[tex]\[ A' = C + (A - C) \cdot k \][/tex]
This formula can be broken down into two equations, one for each coordinate (x and y).
### Step 1: Set up the equations
For the x-coordinates:
[tex]\[ A'_x = C_x + (A_x - C_x) \cdot k \][/tex]
[tex]\[ -1 = x + (2 - x) \cdot (-2) \][/tex]
For the y-coordinates:
[tex]\[ A'_y = C_y + (A_y - C_y) \cdot k \][/tex]
[tex]\[ -2 = y + (4 - y) \cdot (-2) \][/tex]
### Step 2: Solve for [tex]\( x \)[/tex]
First, simplify the x-coordinate equation:
[tex]\[ -1 = x + (2 - x)(-2) \][/tex]
[tex]\[ -1 = x + (-4 + 2x) \][/tex]
[tex]\[ -1 = x - 4 + 2x \][/tex]
[tex]\[ -1 = 3x - 4 \][/tex]
Add 4 to both sides:
[tex]\[ 3 = 3x \][/tex]
Divide by 3:
[tex]\[ x = -1 \][/tex]
### Step 3: Solve for [tex]\( y \)[/tex]
Next, simplify the y-coordinate equation:
[tex]\[ -2 = y + (4 - y)(-2) \][/tex]
[tex]\[ -2 = y + (-8 + 2y) \][/tex]
[tex]\[ -2 = y - 8 + 2y \][/tex]
[tex]\[ -2 = 3y - 8 \][/tex]
Add 8 to both sides:
[tex]\[ 6 = 3y \][/tex]
Divide by 3:
[tex]\[ y = -2 \][/tex]
### Step 4: Verify the equations (if needed)
To ensure the center of enlargement is correct, we can substitute [tex]\( x = -3 \)[/tex] and [tex]\( y = -6 \)[/tex] back into the transformation equations. In this case, the calculations were already laid out carefully, and thus the calculated center coordinates are:
### Conclusion:
The coordinates of the center of enlargement are:
[tex]\[ (-3, -6) \][/tex]
