To determine the image vertices of triangle [tex]\(NMO\)[/tex] after reflecting it over the line [tex]\(x = -5\)[/tex] we will follow these steps:
1. Identify the coordinates of the vertices:
- [tex]\( N(-5, 2) \)[/tex]
- [tex]\( M(-2, 1) \)[/tex]
- [tex]\( O(-3, 3) \)[/tex]
2. Reflect each point over the line [tex]\(x = -5\)[/tex]:
- For a point [tex]\( (x, y) \)[/tex] reflected over the line [tex]\( x = -5 \)[/tex], the new [tex]\( x \)[/tex]-coordinate is determined by:
[tex]\[
x' = 2 \cdot (-5) - x
\][/tex]
while the [tex]\( y \)[/tex]-coordinate remains the same.
3. Calculate the new coordinates for each vertex:
- Reflecting [tex]\( N(-5, 2) \)[/tex]:
[tex]\[
x' = 2 \cdot (-5) - (-5) = -10 + 5 = -5
\][/tex]
Thus, [tex]\( N' = (-5, 2) \)[/tex].
- Reflecting [tex]\( M(-2, 1) \)[/tex]:
[tex]\[
x' = 2 \cdot (-5) - (-2) = -10 + 2 = -8
\][/tex]
Thus, [tex]\( M' = (-8, 1) \)[/tex].
- Reflecting [tex]\( O(-3, 3) \)[/tex]:
[tex]\[
x' = 2 \cdot (-5) - (-3) = -10 + 3 = -7
\][/tex]
Thus, [tex]\( O' = (-7, 3) \)[/tex].
4. Compile the new coordinates:
- [tex]\( N' = (-5, 2) \)[/tex]
- [tex]\( M' = (-8, 1) \)[/tex]
- [tex]\( O' = (-7, 3) \)[/tex]
Therefore, the correct image vertices after reflecting triangle [tex]\(NMO\)[/tex] over the line [tex]\(x = -5\)[/tex] are [tex]\(N'(-5, 2)\)[/tex], [tex]\(M'(-8, 1)\)[/tex], and [tex]\(O'(-7, 3)\)[/tex].
The answer is:
[tex]\[ N'(-5, 2), M'(-8, 1), O'(-7, 3) \][/tex]
