Question

Rotate a point on a line at (−4, 6) 180° counterclockwise. What is the resulting point?

Answer 1

To rotate a point around another point by 180°, you can use the following transformation: If you have a point \( P(x, y) \) and you want to rotate it 180° counterclockwise around the point \( C(h, k) \), the coordinates of the new point \( P' \) after rotation can be found using the formula:

\[
P' = (h - (x - h), k - (y - k))
\]

This simplifies to:

\[
P' = (2h - x, 2k - y)
\]

In this problem, we want to rotate the point \( (−4, 6) \) (which we will consider as the point to be rotated) around the same point \( (−4, 6) \).

Using the formula, we identify \( h = -4 \), \( k = 6 \), \( x = -4 \), and \( y = 6 \).

Substituting into the formula gives:

\[
P' = (2 \cdot -4 - (-4), 2 \cdot 6 - 6)
\]
\[
P' = (-8 + 4, 12 - 6)
\]
\[
P' = (-4, 6)
\]

After a 180° rotation counterclockwise about the point \( (-4, 6) \), the resulting point remains \( (-4, 6) \). Thus, the point remains unchanged:

\[
\text{Resulting point: } (-4, 6)
\]