To find the coordinates of point \( Y \), we can use the formula for the midpoint of a line segment. The midpoint \( M \) of a segment with endpoints \( X(x_1, y_1) \) and \( Y(x_2, y_2) \) is given by:
\[ M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \]
We are given:
- Midpoint \( M = (2, 8) \)
- Point \( X = (10, 4) \)
Let's denote the coordinates of point \( Y \) as \( (x_2, y_2) \). From the midpoint formula, we can set up the following equations:
-
For the x-coordinates: \[ \frac{10 + x_2}{2} = 2 \]
-
For the y-coordinates: \[ \frac{4 + y_2}{2} = 8 \]
Now we can solve these equations.
For the x-coordinate equation: \[ \frac{10 + x_2}{2} = 2 \]
Multiply both sides by 2: \[ 10 + x_2 = 4 \]
Subtract 10 from both sides: \[ x_2 = 4 - 10 \] \[ x_2 = -6 \]
For the y-coordinate equation: \[ \frac{4 + y_2}{2} = 8 \]
Multiply both sides by 2: \[ 4 + y_2 = 16 \]
Subtract 4 from both sides: \[ y_2 = 16 - 4 \] \[ y_2 = 12 \]
Thus, the coordinates of point \( Y \) are:
\[ Y = (-6, 12) \]
Therefore, the ordered pair of point \( Y \) is \( \boxed{(-6, 12)} \).