To solve for the coordinates of point [tex]\( N' \)[/tex] after the dilation, we need to follow these steps:
### Step 1: Understand the Dilation Process
Dilation is a transformation that enlarges or reduces a figure by a scale factor relative to a point called the center of dilation. In this problem, the center of dilation is the origin [tex]\((0, 0)\)[/tex], and the scale factor is given as [tex]\( \frac{2}{5} \)[/tex].
### Step 2: Identify the Original Coordinates of Point [tex]\( N \)[/tex]
From the given choices and the problem statement, we focus on the point [tex]\( N \)[/tex] from the original hexagon, which has coordinates [tex]\( (-2, 2) \)[/tex].
### Step 3: Apply the Scale Factor to the Coordinates
To find the coordinates of [tex]\( N' \)[/tex] after dilation:
- Multiply the [tex]\( x \)[/tex]-coordinate of [tex]\( N \)[/tex] by the scale factor [tex]\( \frac{2}{5} \)[/tex].
- Multiply the [tex]\( y \)[/tex]-coordinate of [tex]\( N \)[/tex] by the scale factor [tex]\( \frac{2}{5} \)[/tex].
Mathematically, if the original point [tex]\( N = (x, y) \)[/tex], then the coordinates of [tex]\( N' \)[/tex] after dilation are [tex]\( ( \frac{2}{5} \times x, \frac{2}{5} \times y ) \)[/tex].
### Step 4: Calculate the New Coordinates
Given point [tex]\( N = (-2, 2) \)[/tex]:
- New [tex]\( x \)[/tex]-coordinate [tex]\( N' = \frac{2}{5} \times (-2) = -\frac{4}{5} = -0.8 \)[/tex]
- New [tex]\( y \)[/tex]-coordinate [tex]\( N' = \frac{2}{5} \times 2 = \frac{4}{5} = 0.8 \)[/tex]
### Step 5: Verify the Solution
The calculations yield the coordinates for point [tex]\( N' \)[/tex] as [tex]\( (-0.8, 0.8) \)[/tex].
### Step 6: Select the Correct Answer
Comparing our calculated coordinates with the given answer choices:
- [tex]\( (-0.4, 0.8) \)[/tex]
- [tex]\( (-0.8, 2.4) \)[/tex]
- [tex]\( (-2.4, 0.8) \)[/tex]
- [tex]\( (-15, 5) \)[/tex]
The correct answer is:
[tex]\[ (-0.8, 0.8) \][/tex]
Therefore, the ordered pair of point [tex]\( N' \)[/tex] is:
[tex]\[ \boxed{(-0.8, 0.8)} \][/tex]
