To find the coordinates of point \( Y \) given the midpoint \( M \) of line segment \( XY \) and point \( X \), we can use the midpoint formula. The midpoint \( M \) of line segment \( XY \) is given by the formula:
\[ M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \]
where \( (x_1, y_1) \) are the coordinates of point \( X \) and \( (x_2, y_2) \) are the coordinates of point \( Y \).
Given:
- Midpoint \( M = (2, 8) \)
- Point \( X = (10, 4) \)
We can set up the equations like this:
\[ \frac{10 + x_2}{2} = 2 \]
\[ \frac{4 + y_2}{2} = 8 \]
Now, let's solve for \( x_2 \) and \( y_2 \).
- For \( x_2 \):
\[ \frac{10 + x_2}{2} = 2 \]
Multiply both sides by 2:
\[ 10 + x_2 = 4 \]
Subtract 10 from both sides:
\[ x_2 = 4 - 10 \]
\[ x_2 = -6 \]
- For \( y_2 \):
\[ \frac{4 + y_2}{2} = 8 \]
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:
\[ Y = (-6, 12) \]
The ordered pair of point \( Y \) is \( (-6, 12) \).