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 \(y = 1\):
-
Calculate the vertical distance from X to the line of reflection: \[ d = y_X - y_{\text{line}} = 9 - 1 = 8 \]
-
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 \(\boxed{(-3, -7)}\).
1 answer