To find the vertices of the reflected image of triangle NMO over the line [tex]\( x = -2 \)[/tex], let's go through the steps of reflection:
1. Identify the coordinates of vertices of triangle NMO:
- [tex]\( N(-5, 2) \)[/tex]
- [tex]\( M(-2, 1) \)[/tex]
- [tex]\( O(-3, 3) \)[/tex]
2. Reflect each vertex over the line [tex]\( x = -2 \)[/tex]:
- For any point [tex]\((x, y)\)[/tex], the reflection over [tex]\( x = a \)[/tex] gives us the new [tex]\( x \)[/tex]-coordinate as [tex]\( 2a - x \)[/tex] and the [tex]\( y \)[/tex]-coordinate remains the same.
3. Calculate the reflected coordinates:
- For [tex]\( N(-5, 2) \)[/tex]:
- The [tex]\( x \)[/tex]-coordinate of the reflection is [tex]\( 2(-2) - (-5) = -4 + 5 = 1 \)[/tex]
- So, [tex]\( N' = (1, 2) \)[/tex]
- For [tex]\( M(-2, 1) \)[/tex]:
- The [tex]\( x \)[/tex]-coordinate remains: [tex]\( 2(-2) - (-2) = -4 + 2 = -2 \)[/tex]
- So, [tex]\( M' = (-2, 1) \)[/tex]
- For [tex]\( O(-3, 3) \)[/tex]:
- The [tex]\( x \)[/tex]-coordinate of the reflection is [tex]\( 2(-2) - (-3) = -4 + 3 = -1 \)[/tex]
- So, [tex]\( O' = (-1, 3) \)[/tex]
By following these steps, we get the reflected vertices:
- [tex]\( N'(1, 2) \)[/tex]
- [tex]\( M'(-2, 1) \)[/tex]
- [tex]\( O'(-1, 3) \)[/tex]
These coordinates match the first option given in the question:
[tex]\[ N'(1, 2), M'(-2, 1), O'(-1, 3) \][/tex]
Therefore, the correct answer is:
[tex]$
\boxed{N^{\prime}(1, 2), M^{\prime}(-2, 1), O^{\prime}(-1, 3)}
$[/tex]
