\( [B] \) If \( \overline{A B} \) is a diameter in the circle \( M \), If \( A(8, Y), B(X, 3) \), \( M(5,7) \) Find the value of \( X+Y \)

To solve for \( X + Y \), we first use the fact that if \( \overline{AB} \) is a diameter of the circle, then point \( M \) (the midpoint of \( \overline{AB} \)) can be calculated using the midpoint formula. The midpoint \( M \) of the line segment connecting points \( A(8, Y) \) and \( B(X, 3) \) is given by: \[ M = \left( \frac{8 + X}{2}, \frac{Y + 3}{2} \right) \] Given that \( M(5, 7) \), we equate the coordinates: 1. For the x-coordinate: \[ \frac{8 + X}{2} = 5 \] Multiplying both sides by 2: \[ 8 + X = 10 \\ X = 10 - 8 = 2 \] 2. For the y-coordinate: \[ \frac{Y + 3}{2} = 7 \] Multiplying both sides by 2: \[ Y + 3 = 14 \\ Y = 14 - 3 = 11 \] Now we can find \( X + Y \): \[ X + Y = 2 + 11 = 13 \] Therefore, the value of \( X + Y \) is \( \boxed{13} \).