Question

Triangle NMO has vertices at N(−5, 2), M(−2, 1), and O(−3 , 3). Determine the vertices of image N′M′O′ if the preimage is reflected over y = −2.

N′(−3, 2), M′(0, 1), O′(−5, 3)
N′(−5, 0), M′(−2, −1), O′(−3, 1)
N′(−5, 1), M′(−2, 0), O′(−3, 2)
N′(−5, −6), M′(−2, −5), O′(−3, −7)

Answer 1

To reflect the points of triangle NMO over the line \( y = -2 \), we first need to understand how reflection works. When we reflect a point \( (x, y) \) over a line \( y = k \), the coordinates of the reflected point \( (x', y') \) can be calculated using the formula:

\[
y' = 2k - y
\]

For our case, the line is \( y = -2 \). So we will replace \( k \) with \(-2\) in the formula:

\[
y' = -4 - y
\]

Now we will apply this reflection to each vertex of triangle NMO.

1. For point **N(−5, 2)**:
\[
y' = -4 - 2 = -6
\]
Thus, \( N' = (-5, -6) \).

2. For point **M(−2, 1)**:
\[
y' = -4 - 1 = -5
\]
Thus, \( M' = (-2, -5) \).

3. For point **O(−3, 3)**:
\[
y' = -4 - 3 = -7
\]
Thus, \( O' = (-3, -7) \).

Putting it all together, the vertices of the reflected triangle N'M'O' are:

\[
N'(-5, -6), M'(-2, -5), O'(-3, -7)
\]

Thus, the correct answer is:

\[
N'(-5, -6), M'(-2, -5), O'(-3, -7)
\]

This matches with the choice:

N′(−5, −6), M′(−2, −5), O′(−3, −7).