Question

Triangle ABC has been reflected over a vertical line of reflection. The coordinates of vertices A, B, C, and B' are shown. What are the coordinates of vertex A'?
Point A: (0, 9)
Point B: (-5, 5)
Point C: (-5, -1)
Point B': (11, 5)

Answer 1

To find the coordinates of vertex A' after triangle ABC has been reflected over a vertical line, we first need to determine the line of reflection. We know that point B is reflected to point B'.

The reflection of point B(-5, 5) to point B' (11, 5) occurs horizontally. Therefore, the vertical line of reflection must be midway between points B and B'.

1. Calculate the midpoint (M) between B and B':
\[
M_x = \frac{B_x + B'_x}{2} = \frac{-5 + 11}{2} = \frac{6}{2} = 3
\]
\[
M_y = \frac{B_y + B'_y}{2} = \frac{5 + 5}{2} = 5
\]
So the midpoint M is (3, 5).

2. The line of reflection has the x-coordinate of 3. This means it is the line \( x = 3 \).

3. To find A' from A, we apply the reflection across this line:
- The x-coordinate of A (0, 9) is reflected over the line \( x = 3 \).
- The distance from A to the line (3) is:
\[
\text{Distance} = 3 - 0 = 3
\]
- Since reflections are equal distances on both sides of the line, the x-coordinate of A' would be:
\[
A'_x = 3 + \text{Distance} = 3 + 3 = 6
\]
- The y-coordinate remains unchanged since it is a vertical reflection. Thus, the y-coordinate of A' is the same as A's:
\[
A'_y = 9
\]

4. Therefore, the coordinates of vertex A' are \( \boxed{(6, 9)} \).