Question

Triangle XYZ is reflected over a horizontal line of reflection to create triangle X'Y'Z'. What are the coordinates of vertex X'?
Triangle XYZ coordinates: X(-3,9), Y(-2,4), Z(3,4)
Triangle X'Y'Z' coordinate given: Z(3,-2)

Answer 1

When a point is reflected over a horizontal line, the x-coordinate remains the same, and the y-coordinate is transformed based on the line of reflection.

First, let's find the horizontal line of reflection using the given coordinates of vertex Z in triangle XYZ and Z' in triangle X'Y'Z'. The coordinates of Z are (3, 4) and Z' are (3, -2).

The line of reflection must be equidistant from points Z and Z'. The y-coordinate of Z is 4, and that of Z' is -2.

The y-coordinate of the line of reflection can be found using the midpoint formula:

\[
y_{\text{mid}} = \frac{y_Z + y_{Z'}}{2} = \frac{4 + (-2)}{2} = \frac{2}{2} = 1
\]

Thus, the line of reflection is the horizontal line \(y = 1\).

Now, to find the coordinates of vertex X' after the reflection, we take the coordinates of vertex X, which are (-3, 9). The y-coordinate reflects in relation to the line \(y = 1\):

1. Calculate the vertical distance from X to the line of reflection:
\[
d = y_X - y_{\text{line}} = 9 - 1 = 8
\]

2. To find the y-coordinate of X', we subtract this distance from the line of reflection:
\[
y_{X'} = y_{\text{line}} - d = 1 - 8 = -7
\]

The x-coordinate remains the same since it is a horizontal reflection, therefore:

The coordinates of vertex X' are:
\[
X' = (-3, -7)
\]

Thus, the coordinates of vertex X' are \(\boxed{(-3, -7)}\).