Question

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

The coordinates of vertex A' are (_, _)

Answer 1

To find the coordinates of vertex A' after triangle ABC has been reflected over a vertical line, we need to determine the equation of that vertical line. We know the coordinates of vertex B' (the reflected point of B) are (11, 5).

Vertex B is at B(-5, 5). The reflection of a point across a vertical line can be found by measuring the horizontal distance to the line of reflection and then adding that distance in the opposite direction.

1. Find the horizontal distance from B to the line of reflection:
- The x-coordinate of point B is -5. The x-coordinate of point B' is 11.
- The midpoint between B and B' will lie on the line of reflection.
- The midpoint's x-coordinate is calculated as:
\[
\frac{-5 + 11}{2} = \frac{6}{2} = 3
\]
Therefore, the line of reflection is the vertical line \( x = 3 \).

2. Now, we need to find the reflected coordinates of point A(0, 9). The x-coordinate of A is 0:
- The distance from A to the line of reflection is:
\[
|0 - 3| = 3
\]
- To find the reflected x-coordinate A', we subtract this distance from the line of reflection:
\[
3 + 3 = 6
\]
So, the x-coordinate of A' is 6.

3. The y-coordinate remains unchanged during a reflection over a vertical line. Therefore, the y-coordinate of A' is the same as that of A:
- The y-coordinate of A is 9.

Thus, the coordinates of vertex A' are \( (6, 9) \).

Put together, the coordinates of vertex A' are:
\[
(6, 9)
\]