To find the reflection that maps points B and C given the points A(-4, 2) and A'(2, -4), we will proceed with the following steps:
1. Identify the midpoint between A and A':
Since the reflection maps A to A’, the midpoint of A and A' is the line of reflection.
- The coordinates of A are (-4, 2).
- The coordinates of A' are (2, -4).
Calculate the midpoint (M) of point A and point A':
\(\[
M_x = \frac{-4 + 2}{2} = \frac{-2}{2} = -1
\]\)
\(\[
M_y = \frac{2 + (-4)}{2} = \frac{-2}{2} = -1
\]\)
Thus, the midpoint \(\( M = (-1, -1) \)\).
2. Reflect point B (5, 5):
Using the midpoint (line of reflection) and point B, we can determine the reflection of B, denoted as B'.
- Reflect the x-coordinate of B:
\(\[
B'_x = 2 \cdot M_x - B_x = 2 \cdot (-1) - 5 = -2 - 5 = -7
\]\)
- Reflect the y-coordinate of B:
\(\[
B'_y = 2 \cdot M_y - B_y = 2 \cdot (-1) - 5 = -2 - 5 = -7
\]\)
Therefore, the reflection of point B, or \(\( B' \)\), is \(\((-7, -7)\)\).
3. Reflect point C (1, 6):
Similarly, using the same midpoint (line of reflection) and point C, we can find the reflection of C, denoted as C'.
- Reflect the x-coordinate of C:
\(\[
C'_x = 2 \cdot M_x - C_x = 2 \cdot (-1) - 1 = -2 - 1 = -3
\]\)
- Reflect the y-coordinate of C:
\(\[
C'_y = 2 \cdot M_y - C_y = 2 \cdot (-1) - 6 = -2 - 6 = -8
\]\)
Therefore, the reflection of point C, or \(\( C' \)\), is \(\((-3, -8)\)\).
In conclusion, the coordinates of B' and C' after being mapped by the same reflection that mapped A to A' are:
- \(\( B' = (-7, -7) \)\)
- \(\( C' = (-3, -8) \)\)
