To find the length of \( XY \) in the sector \( AOB \) with angle \( AOB = 60^\circ \) and radius \( OA = OB = 12 \), we can follow these steps:
Step 1: Find the length of \( AB \)
Since \( AOB \) is a sector with \( \angle AOB = 60^\circ \), we can find the arc length \( AB \) using the formula for the chord length in a circle:
\[ AB = OA \cdot \sin\left(\frac{\angle AOB}{2}\right). \]
The half angle is:
\[ \frac{\angle AOB}{2} = \frac{60^\circ}{2} = 30^\circ. \]
Then, the length of \( AB \) is:
\[ AB = 12 \cdot \sin(30^\circ) = 12 \cdot \frac{1}{2} = 6. \]
Step 2: Find the coordinates of points \( A \) and \( B \)
We can place \( O \) at the origin of a coordinate system. Thus, using polar coordinates:
- Point \( A \) (at \( \angle -30^\circ \)) has coordinates:
\[ A = \left(12 \cos(-30^\circ), 12 \sin(-30^\circ)\right) = \left(12 \cdot \frac{\sqrt{3}}{2}, 12 \cdot \left(-\frac{1}{2}\right)\right) = \left(6\sqrt{3}, -6\right). \]
- Point \( B \) (at \( \angle 30^\circ \)) has coordinates:
\[ B = \left(12 \cos(30^\circ), 12 \sin(30^\circ)\right) = \left(12 \cdot \frac{\sqrt{3}}{2}, 12 \cdot \frac{1}{2}\right) = \left(6\sqrt{3}, 6\right). \]
Step 3: Find the coordinates of point \( X \)
The line \( OY \) is a vertical line through the center \( O(0, 0) \). The point \( Y \) is the midpoint of the chord \( AB \). To find the coordinates of point \( X \), we will average the y-coordinates of points \( A \) and \( B \):
\[ Y_x = \frac{A_x + B_x}{2} = \frac{(6\sqrt{3} + 6\sqrt{3})}{2} = 6\sqrt{3} \quad \text{(x-coordinate)}, \]
\[ Y_y = \frac{A_y + B_y}{2} = \frac{(-6 + 6)}{2} = 0 \quad \text{(y-coordinate)}. \]
So point \( Y \) is at \( (6\sqrt{3}, 0) \).
Step 4: Find the length of \( XY \)
Point \( X \) is on the horizontal line segment \( AB \) and point \( Y \) is on the vertical line \( OY \). Since \( X \) is at the midpoint of \( AB \), we have already established that:
\[ X = (6\sqrt{3}, 0), \]
and since both points \( Y \) and \( X \) have the same coordinates at \( x = 6\sqrt{3} \):
The length \( XY \) is determined by the vertical distance from point \( Y(6\sqrt{3}, 0) \) to point \( X(6\sqrt{3}, -6) \):
\[ XY = |Y_y - X_y| = |0 - (-6)| = |0 + 6| = 6. \]
Final Answer:
Thus, the length of \( XY \) is
\[ \boxed{6}. \]